A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

A SCALED TW-PINN: A PHYSICS-INF ORMED NEURAL NETW ORK F OR TRA VELING W A VE SOLUTIONS OF REA CTION-DIFFUSION EQUA TIONS WITH GENERAL COEFFICIENTS SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG Abstract. W e propose an efficient and generalizable physics-informed neu- ral netw ork (PINN) framew ork for computing trav eling wa ve solutions of n - dimensional reaction-diffusion equations with v arious reaction and diffusion co- efficients. By applying a scaling transformation with the trav eling w av e form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coe fficients. This reduction leads to the prop osed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for differen t co efficient c hoices and spatial dimensions. W e also prov e a universal appro ximation prop erty of the proposed PINN solv er for tra v eling w av e solutions. Numerical experiments in one and t wo dimensions, together with a comparison to the existing wav e- PINN method, demonstrate the accuracy , flexibility , and sup erior performance of scaled TW-PINN. Finally , w e explore an extension of the framework to the Fisher’s equation with general initial conditions. 1. Introduction Reaction-diffusion systems constitute a fundamental class of partial differen tial equations (PDEs), c haracterizing the interpla y b etw een the lo cal reaction kinetics and the diffusive transp ort. They pla y a central role in physics, chemistry , biology , and ecology , providing a unifying framew ork for a wide range of phenomena suc h as excitable media [6, 29], catalytic surface reactions [12, 5], neuroscience [11, 9], and p opulation dynamics [8, 25]. Their broad applicability stems from their capability to capture the core mechanisms underlying diverse spatiotemp oral processes in natural systems. The general reaction-diffusion system takes the form ∂ t u = ∇ · ( D ∇ u ) + R ( u ) , where D is a diagonal matrix of diffusion coefficients and R ( u ) denotes a nonlinear reaction term, t ypically inv olving reaction co efficients. Dep ending on the reaction term and the diffusion co efficients, the reaction-diffusion system exhibits a v ariety of qualitatively distinct solution behaviors. These include tra v eling wa ves [8, 27, 31], spiral/scroll wa ves [29, 30, 14], pattern formation [26, 21, 20] and multistable phenomena [13, 1, 7]. Within each class, k ey qualitative features of the solutions, including spatial profiles, characteristic length scales, and steepness, are gov erned 2020 Mathematics Subje ct Classific ation. 35K57, 68T07. Key wor ds and phr ases. Ph ysics-informed neural network, Reaction-diffusion equation, T rav- eling wa ve solution, Scaling transformation, W av e speed, Universal appro ximation property . 1 2 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG b y the diffusion co efficients in D and the parameters in R ( u ), particularly reaction co efficien ts. Ph ysics-informed neural netw orks (PINNs) [22] hav e recen tly b ecome a widely used machine learning approach for approximating the solutions of PDEs. How ever, PINNs often struggle to resolve solutions with sharp transitions [16, 28], a limita- tion c haracterized b y slo w con v ergence and inefficien t learning. In resp onse to this limitation, some PINN v ariants hav e been prop osed for discon tin uous and sharp solutions [18, 17, 15]. In [24], the wa ve-PINN metho d was prop osed b y introduc- ing a residual weigh ting scheme to improv e the PINN appro ximation of the sharp tra veling wa v e in Fisher’s equation with large reaction co efficien ts. Despite this impro vemen t, when the reaction co efficient b ecomes extremely large, wa v e-PINN fails to capture the wa v e front accurately , esp ecially with resp ect to the predicted w av e sp eed, as shown later in Section 5.3. Therefore, it remains challenging to ac- curately compute the sharp trav eling wa ve front of the reaction-diffusion equation with PINNs when the reaction co efficien t is large. T o address this issue of resolving the sharp w a v e front in the n -dimensional reaction-diffusion equation, we first apply a scaling transformation that normalizes the reaction and diffusion co efficien ts to unit y . In the resulting scaled equation, the tra veling w av e profile b ecomes less steep, making the problem more suitable for PINN training. By further exploiting the trav eling wa ve form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation. With this reduction, even a PINN with simple architecture can b e trained efficien tly on the scaled equation. The prop osed PINN solv er thus consists only of a wa ve la y er, a single hidden la y er, and an output lay er. The w av e la y er con tains a trainable parameter corresp onding to the predicted wa v e sp eed, enabling direct monitoring of wa ve sp eed learning during training and serv es as an indicator of whether a training run ac hiev es physical conv ergence. Once the PINN solver is trained on the one-dimensional scaled equation, w e com bine it with the scaling and inv erse transformations to construct a scaling PINN framework, referred to as scaled TW- PINN, for computing tra veling w av e solutions of n -dimensional reaction-diffusion equations with differen t reaction and diffusion co efficients. Numerical experiments sho w that our scaled TW-PINN accurately captures the sharp wa v e front. The outline of this pap er is as follo ws. In Section 2, w e in tro duce the isotropic scalar reaction-diffusion equation, present the scaling transformation, and review the dimension-independent tra v eling w a v e reduction. W e also summarize four rep- resen tative reaction terms, each with a corresponding exact tra veling wa ve solution and asso ciated sp ecial wa ve sp eed. Section 3 prop oses the PINN solver, including its architecture, loss functions, and training configuration with conv ergence analy- sis, follow ed by the solution pip eline of the ov erall scaling PINN framework. It is sho wn that the prop osed PINN solv er has a universal approximation prop erty for tra veling wa ve solutions in Section 4. W e pro vide n umerical experiments in one and t wo dimensions, together with a comparison to the existing wa v e-PINN metho d, in Section 5. Section 6 further explores an extension of the framework to Fisher’s equation with general initial conditions. The concluding section giv es some remarks and outlines future developmen ts. A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 3 2. Scaling and tra veling w a ve form of reaction-diffusion equa tions This section introduces the reaction-diffusion equation and its dimension-independent tra veling w av e reduction. A key step from the original equation to its reduced form is the scaling transformation, which remov es the explicit dep endence on the reac- tion and diffusion coefficients and, more imp ortantly , decreases the sharpness of the w av e front. As a result, the scaled equation is b etter suited for PINN training. 2.1. Reaction-diffusion equations. Consider the isotropic scalar reaction-diffusion equation, in which the diffusion co efficient is iden tical in all spatial directions. The go verning equation tak es the form (1) ∂ t u ( x , t ) = D ∇ 2 x u ( x , t ) + R ( u ( x , t )) , where x = ( x 1 , · · · , x n ) with n the spatial dimension, D > 0 denotes the diffusion co efficien t and R ( u ) represen ts the reaction term. The reaction term is assumed to ha ve the general form [10] R ( u ) = ρ u p (1 − u q )( u − a ) r , where p, q > 0, r ∈ { 0 , 1 } , and a ∈ (0 , 1). The exp onen t r = 0 corresp onds to the monostable nonlinearity , while r = 1 yields the classical bistable nonlinearity . The parameter ρ > 0 is the reaction co efficient that con trols the ov erall strength of the nonlinear kinetics. In (1), increasing ρ accelerates the lo cal gro wth or decay of u and hence shortens the c haracteristic reaction time scale. Conv ersely , a decrease in ρ reverses this trend. 2.2. Scaling. W e apply a scaling transformation to (1) b y writing (2) τ = ρ t, ξ = r ρ D x , v ( ξ , τ ) = u ( x , t ) . Then (1) b ecomes (3) ∂ τ v = ∇ 2 ξ v + v p (1 − v q )( v − a ) r , where both the reaction and diffusion coefficients b ecome one. It is easy to recov er the original v ariables b y the inv erse transformation, (4) t = 1 ρ τ , x = s D ρ ξ , u ( x , t ) = v ( ξ , τ ) . Consequen tly , once the scaled equation (3) is solved, the solution of the original equation (1) is obtained by reco v ering u from the scaled solution v . 2.3. Dimension-indep endent tra v eling wa ve reduction. A distinguished class of solutions to (3) is giv en by tra veling w av es of the form (5) v ( ξ , τ ) = V ( ζ ) , ζ = n · ξ − cτ , where ζ is the analytical trav eling wa ve co ordinate, n denotes a unit v ector sp ec- ifying the direction of propagation and c is the wa v e sp eed. Under the tra v eling w av e form (5), the scaled reaction-diffusion equation (3) reduces to the ordinary differen tial equation (ODE), (6) V ′′ + cV ′ + V p (1 − V q )( V − a ) r = 0 , where primes denote differentiation with resp ect to ζ . This reduction sho ws that, regardless of the spatial dimension, the trav eling wa v e form leads to the same 4 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG second-order ODE for the solution V . As a result, the trav eling wa v e solution V , as well as the sp eed c , is the same in an y spatial dimension. Th us the trav eling w av e form is in trinsically one-dimensional, even in higher dimensions. This prop erty justifies the use of a single PINN solver for computing trav eling wa v e solutions of the n -dimensional reaction-diffusion equation. 2.4. Reaction terms and corresp onding exact tra veling wa ve solutions. W e study four representativ e reaction-diffusion equations that p ossess tra v eling w av e solutions: Fisher’s, Newell-Whitehead-Segel (NWS), Zeldovic h, and bistable equations. Eac h equation is characterized by a sp ecific nonlinear reaction term R ( u ) listed in T able 1. It is w orth noting that the NWS equation with q = 2 is the Allen-Cahn equation. The spatially homogeneous equilibrium states are determined solely by the reaction term R ( u ). A t ypical profile of the trav eling wa v e solution V is monotone and connects tw o equilibrium states. Sp ecifically , V ( ζ ) satisfies (7) lim ζ →−∞ V ( ζ ) = v − , lim ζ →∞ V ( ζ ) = v + , so that the trav eling wa v e represen ts a transition from the equilibrium state v − to the other equilibrium state v + . F or the Fisher’s, NWS, and Zeldovic h equations, the equilibrium states are ( v − , v + ) = (1 , 0), whereas for the bistable equation, they are ( v − , v + ) = ( a, 1). T able 1. Reaction terms R ( u ) for Fisher’s, NWS, Zeldovic h, and bistable equations. R ( u ) P arameter range Fisher ρ u (1 − u ) – NWS ρ u (1 − u q ) q > 0 Zeldo vich ρ u 2 (1 − u ) – bistable ρ u (1 − u )( u − a ) 0 < a < 1 All four equations admit the exact tra veling wa ve solution in closed form with a sp ecial w a v e sp eed c , as summarized in T able 2. These explicit solutions reveal that the steepness of the w av e front scales prop ortionally to p ρ/D . In particular, large v alues of ρ/D lead to the thin transition lay er with pronounced gradients, which p oses significan t c hallenges for b oth traditional n umerical metho ds [10] and neural net work approaches [24]. 3. Scaling PINN framework for tra veling w a ve solutions Using the scaling transformation (2), and the tra veling wa ve co ordinate (5) for the tra v eling w a v e solution, we can simplify the task of solving a class of n - dimensional reaction-diffusion equations (1) with the same reaction term but dif- feren t co efficients to the scaled one-dimensional reaction-diffusion equation, (8) ∂ τ v = ∂ ξξ v + v p (1 − v q )( v − a ) r . T o compute the trav eling wa ve solution to this reaction-diffusion equation (8), we in tro duce a physics-informed neural net w ork (PINN) solver. A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 5 T able 2. Sp ecial wa v e speed c and its corresponding closed-form tra veling wa ve solution u ( x , t ) for Fisher’s, NWS, Zeldo vic h, and bistable equations. c u ( x , t ) Fisher 5 q ρD 6 1 { 1+exp [ √ ρ 6 D ( n · x − ct ) ] } 2 NWS q +4 √ 2 q +4 √ ρD n 1 2 + 1 2 tanh h − q 2 √ 2 q +4 p ρ D ( n · x − ct ) io 2 q Zeldo vich q ρD 2 1 1+exp [ √ ρ 2 D ( n · x − ct ) ] bistable − (1 + a ) q ρD 2 1+ a 2 + 1 − a 2 tanh  1 − a 4 q 2 ρ D ( n · x − ct )  Training Wave layer Output layer Hidden layer Figure 1. Schematic of the PINN architecture. 3.1. Arc hitecture. The prop osed solver adopts a PINN architecture, illustrated in Fig. 1. The arc hitecture consists of three lay ers: a w av e la y er [3, 24] that imp oses the trav eling wa ve form, a single hidden lay er resp onsible for function appro ximation in accordance with the universal appro ximation theorem, and an output lay er that enforces the equilibrium states. The net work takes ( ξ , τ ) as input. Motiv ated by (5), the wa v e la y er introduces the predicted trav eling w a ve co ordinate in one-dimensional form, (9) ˆ ζ = ξ − ω τ , where ω is a trainable parameter representing the predicted wa v e sp eed. This parameter allo ws the predicted wa v e speed to b e monitored directly during training and is used in Section 3.3 as a diagnostic for ph ysical conv ergence. It should b e noted that the wa ve la y er used for training here is extended to multiple spatial dimensions in (12) emplo yed for the scaling PINN framework in Section 3.4. The hidden lay er is implemented as a fully connected lay er with N neurons, pro ducing the output h ( ˆ ζ ) = N X i =1 c i σ ( a i ˆ ζ + b i ) , where σ denotes a sigmoid activ ation function, defined as an y bounded, differen- tiable real-v alued function on R with strictly positive deriv ativ e. In this paper, our 6 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG net work employs the logistic sigmoid function for σ . T o ensure that the solution remains within the equilibrium states ( v − , v + ), an output constrain t is imp osed. Since the trav eling wa v e solution lies in the interv al ( v − , v + ), w e introduce the constrain t function ϕ , defined by (10) ϕ ( s ) = v − + ( v + − v − ) 1 1 + exp ( − s ) , s ∈ R . This construction guarantees that ϕ ( s ) ∈ ( v − , v + ) for all s ∈ R , whic h eliminates spurious oscillations around the wa ve front. Moreov er, as ϕ is monotone, the net- w ork inherits this monotonicity . The final net work output is then given b y (11) ˆ v ( ξ , τ ) = ϕ N X i =1 c i σ ( a i ˆ ζ + b i ) ! . In our proposed net work, only the predicted wa v e sp eed ω and the w eights of the hidden lay er, indicated in yello w in Fig. 1, are trainable, resulting in a compact and structurally constrained architecture. As sho wn in Section 4, this netw ork has the univ ersal appro ximation property for trav eling w a v e solutions while preserving the prescrib ed equilibrium states. 3.2. Loss functions. F rom (8), the reaction-diffusion equation can b e rewritten as ∂ τ v + N [ v ] = 0 , N [ v ] = − ∂ ξξ − v p (1 − v q )( v − a ) r , sub ject to the initial condition v ( ξ , 0) = v 0 ( ξ ) and some b oundary conditions. In the PINN solver, the go v erning equation and data constraints are incorp orated directly in to the loss function. This approac h restricts the appro ximation space so that the learned solution satisfies the equation together with the prescrib ed conditions. F ollowing [22], the loss is given b y L = L ICBC + L r . where L ICBC = 1 N ICBC N ICBC X i =1   ˆ v ( ξ i v , τ i v ) − v i   2 , L r = 1 N r N r X i =1   ˆ v τ ( ξ i r , τ i r ) + N  ˆ v ( ξ i r , τ i r )    2 . Here, { ξ i v , τ i v , v i } N ICBC i =1 denote the initial and b oundary training data on v ( ξ , τ ) and { ξ i r , τ i r } N r i =1 sp ecify the collo cations p oints for equation residual. 3.3. T raining configuration and conv ergence analysis. The training ( ξ , τ )- domain for the scaled reaction-diffusion equation (8) is [ − 5000 , 5000] × [0 , 2000]. W e use N ICBC = 1024 collo cation p oints for the initial and b oundary loss L ICBC , and N r = 1024 collo cation p oints for the residual loss L r . All collo cation p oints are generated via Latin h yp ercub e sampling. The netw ork parameters are initialized with different random seeds. T raining is p erformed for 100 , 000 ep o c hs using the Adam optimizer with an initial learning rate of 0 . 01, together with cosine annealing to gradually decrease the learning rate throughout the training pro cess. Under the configuration sp ecified abov e, most training runs, eac h utilizing a dif- feren t random seed initialization, ac hiev e ph ysical conv ergence, while a few exhibit A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 7 spurious con vergence. Figure 2 sho ws representativ e ph ysical and spurious con v er- gen t runs for Fisher’s equation by showing the evolution of the training loss L and the predicted w a v e speed ω . Similar phenomena occur for the NWS, Zeldo vic h, and bistable equations; Fisher’s equation is presented as a representativ e example. In 0 20000 40000 60000 80000 100000 epochs 1 0 1 0 1 0 8 1 0 6 1 0 4 1 0 2 1 0 0 Physical Spurious 0 20000 40000 60000 80000 100000 epochs 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Physical Spurious Exact Figure 2. T raining b ehavior on the ph ysical (blue) and spurious (orange) con vergence for Fisher’s equation: log-scale training loss L (left) and predicted wa ve sp eed ω (right) compared to the exact w av e sp eed (dashed). the ph ysically con v ergen t case, the training loss decays steadily tow ard 10 − 10 with only minor fluctuations. In the mean while, the predicted wa ve sp eed ω exhibits oscillations of decreasing amplitude and gradually approac hes the exact w a v e sp eed c = 5 √ 6 ≈ 2 . 04. In contrast, for spuriously conv ergent runs, the loss oscillates p er- sisten tly within the range of [10 − 9 , 10 − 4 ] without stabilization, and the predicted w av e sp eed remains separated from the exact wa ve sp eed. This qualitativ e differ- ence suggests that accurate identification of the wa ve sp eed pla ys an imp ortant role in ph ysical conv ergence. F or eac h equation, training is therefore rep eated with differen t random seeds until ten ph ysical con v ergent solv ers are obtained. Those con vergen t solvers are used for the numerical experiments in Section 5. T able 3 rep orts the discrepancy b etw een the exact wa v e sp eed c in (5) and the predicted w av e sp eed ω in (9) for Fisher’s, NWS ( q = 2), Zeldovic h, and bistable ( a = 0 . 2) equations. F or eac h case, the mean absolute error | c − ω | is computed from those ten ph ysically conv ergent runs, and the corresp onding standard deviation is shown in parentheses. While Fisher’s, Zeldo vic h, and bistable equations yield consisten tly small errors, the NWS equation exhibits comparativ ely large errors. W e also ob- T able 3. Absolute error in predicted w av e sp eed on original and restricted domains. Fisher NWS ( q = 2) Zeldovic h bistable ( a = 0 . 2) | c − ω | original 1.31e-4 (8.82e-5) 2.94e-4 (2.39e-4) 4.14e-5 (2.89e-5) 4.23e-5 (4.95e-5) restricted 1.91e-6 (1.54e-6) 1.98e-6 (1.16e-6) 1.73e-6 (2.24e-6) 1.09e-6 (6.92e-7) serv e that the physical conv ergence behavior v aries across equations. In particular, Fisher’s, Zeldovic h, and bistable equations predict the w a v e sp eed accurately under most random initializations, whereas the NWS equation frequently fails to predict correctly . 8 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG W e further take a lo ok at the case of spurious conv ergence. The trav eling wa ve solution connects the equilibrium states and propagates through the domain, so ac- curate learning of the w av e speed requires sufficient resolution near the w av e fron t. Ho wev er, the original training ( ξ , τ )-domain is to o large ([ − 5000 , 5000] × [0 , 2000]) relativ e to the num b er of collo cation points ( N ICBC = 1024 and N r = 1024), re- sulting in a sparse sampling of the wa v e front region. T o improv e resolution, tw o approac hes are considered: increasing the num b er of collo cation p oin ts or restricting the training domain while assuming that outer regions remain close to the equilib- rium states. Empirical observ ations indicate that domain restriction is significantly more effective. W e restrict the training ( ξ , τ )-domain to [ − 500 , 500] × [0 , 20]. This adjustmen t increases the effective sampling density near the w a v e fron t and concen- trates training on the dynamically relev ant region. Figure 3 compares training on the original and restricted domains using the random seed, whic h fails to con v erge ph ysically on the original domain. On the original domain, the PINN solver fails to 0 20000 40000 60000 80000 100000 epochs 1 0 9 1 0 7 1 0 5 1 0 3 1 0 1 Original R estricted 0 20000 40000 60000 80000 100000 epochs 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Original R estricted Exact Figure 3. T raining behavior on the original (orange) and re- stricted (green) domains for Fisher’s equation: log-scale training loss L (left) and predicted wa ve sp eed ω (right) compared to the exact wa ve speed (dashed). learn the correct wa ve sp eed, and the loss oscillates p ersistently within the range of [10 − 9 , 10 − 4 ] without stabilization, corresp onding to the case of spurious conv er- gence. On the restricted domain, the predicted wa ve speed conv erges rapidly to the exact w av e speed without oscillations, and the loss decreases smoothly , which corre- sp onds to the ph ysical conv ergence. This b ehavior is observed for all tested random seeds. These results show that increasing the effective sampling density near the w av e fron t is crucial for accurate w av e sp eed identification and stable training. W e also notice that ph ysical con vergence is faster than in the setting of the original do- main. F or consistency , training is rep eated using the same random seeds as in the original domain, which yielded con vergen t physical solutions. T able 3 also rep orts the discrepancy b etw een the exact wa ve sp eed c and the corresp onding predicted w av e sp eed ω on restricted domains. A substantial improv emen t is observed on the restricted domain. The impro ved accuracy of the appro ximation b y the scaling PINN framew ork on the restricted domain is confirmed by the n umerical results in Section 5. In addition, separate solvers are trained for the NWS equation on the restricted domain with q = 3 and q = 4, resp e ctiv ely . As seen in T able 4, the errors in the predicted w av e speed remain on the order of 10 − 6 . These results demonstrate that PINN accurately learns the w a ve sp eed even for q ≥ 2. This robust p erformance A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 9 stands in contrast to WENO schemes, which sho ws the difficulty in accurately capturing the wa ve sp eed [10]. T able 4. Absolute error in the predicted wa ve sp eed for NWS equation ( q = 3 , 4) on the restricted domain. q 3 4 | c − ω | 1.50e-6 3.63e-6 3.4. Solution pip eline. After training the PINN solver in Section 3.3, w e incorp o- rate it into the prop osed scaling PINN framework to compute numerical solutions of the reaction-diffusion equation (1). Figure 4 illustrates the solution pip eline of the framew ork. The pro cedure consists of three stages: a scaling transformation, appro ximation of the scaled equation using the trained PINN solv er, and an inv erse transformation. Figure 4. Solution pip eline of the prop osed scaling PINN frame- w ork: a scaling transformation, appro ximation of the scaled equa- tion using a trained PINN solv er, and an inv erse transformation. The framework takes ( n , x , t ) as input. The spatial v ariable x and temp oral v ariable t are first pro cessed b y a pre-processing lay er that performs the scaling transformation (2), giving the scaled v ariables ξ and τ . Corresp ondingly , the orig- inal reaction-diffusion equation is transformed into the scaled form in which the reaction and diffusion co efficien ts are rescaled to unity . The resulting v ariables ( n , ξ , τ ) are then passed through the wa v e lay er, (12) ˆ ζ = n · ξ − ω τ , with the learned wa ve sp eed ω . This lay er op erates in a dimensionally independent manner, motiv ated by the observ ation in Section 2.3 that the trav eling wa v e solution is indep endent of the spatial dimension. The predicted tra veling wa v e co ordinate ˆ ζ is then passed through the single hidden la yer and the output constraint to compute the scaled solution ˆ v b y the trained PINN solv er. This step corresponds to solving the scaled equation using the prop osed PINN solver, represented by the central blo c k in Fig. 4. Finally , the approximate solution ˆ v is transformed bac k to ˆ u through the in v erse transformation (4). This step reco v ers the solution of the original reaction-diffusion equation with the given reaction and diffusion co efficients. Since the scaling is linear, the in verse transformation is also linear and can therefore b e computed efficiently . This pip eline enables a single trained solver to consisten tly compute solutions for a wide range of coefficients and arbitrary spatial dimensions within a unified framework. 10 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG 4. Universal approxima tion proper ty for tra veling w a ve solutions In this section, we show that the arc hitecture prop osed in Section 3.1 p ossesses the universal appro ximation prop ert y for trav eling w av e solutions. Theorem 4.1 (Univ ersal approximation theorem [4]) . L et σ b e any c ontinuous discriminatory function. Then finite sums of the form G ( ζ ) = N X i =1 c i σ ( a T i ζ + b i ) ar e dense in C ( I n ) . In other wor ds, given any F ∈ C ( I n ) and ε > 0 , ther e is a sum, G ( ζ ) , of ab ove form, for which | F ( ζ ) − G ( ζ ) | < ε for al l ζ ∈ I n . R emark 4.2 . In the abov e theorem, I n = [0 , 1] n ⊂ R n denotes the n -dimensional unit cub e, and a T i ζ is the inner pro duct of a i ∈ R n and ζ ∈ I n . The universal appro ximation result stated on C ( I n ) extends to C ( K ) for an y compact set K ⊂ R n b y composing F with an affine bijection b etw een K and a subset of I n . A function σ is said to be discriminatory if, for a measure µ ∈ M ( I n ), the space of finite signed regular Borel measures on I n , Z I n σ ( a T ζ + b ) d µ ( ζ ) = 0 , for all a ∈ R n and b ∈ R implies that µ = 0. The activ ation function employ ed in Section 3.1 satisfies the con tin uity and discriminatory assumptions of Theorem 4.1. Therefore, the corresp onding shallow net works are dense in C ( K ) for each compact set K ⊂ R n . The follo wing lemma sho ws that this approximation prop erty remains v alid for one-dimensional functions under comp osition with a constraint function. Lemma 4.3 (Univ ersal approximation under constrain ts) . L et K ⊂ R b e a c omp act set, and V : K → ( α, β ) a c ontinuous function. Supp ose σ is a sigmoid function, and ϕ : R → ( α, β ) is c ontinuous, surje ctive, and strictly monotone. Then for any ε > 0 , ther e exist an inte ger N ∈ N and c onstants a i , b i , c i ∈ R for i = 1 , . . . , N such that for al l ζ ∈ K ,      V ( ζ ) − ϕ N X i =1 c i σ ( a i ζ + b i ) !      < ε. Pr o of. Since ϕ is con tin uous and bijective, its inv erse ϕ − 1 exists and is con tinuous. Define F ( ζ ) = ϕ − 1 ( V ( ζ )). Since V is contin uous on K , the comp osition F = ϕ − 1 ◦ V is con tin uous on K . Since K is compact, the image F ( K ) is compact and th us b ounded. Let I F b e a compact interv al such that F ( K ) is con tained in the interior of I F . The con tin uit y of ϕ on the compact set I F implies the uniform con tinuit y of ϕ | I F . Fix ε > 0. By the uniform contin uity of ϕ | I F , there exists δ > 0 independent of ζ ∈ K such that ( F ( ζ ) − δ, F ( ζ ) + δ ) ⊂ I F and if | F ( ζ ) − y | < δ , then | ϕ ( F ( ζ )) − ϕ ( y ) | < ε, for all ζ ∈ K and y ∈ I F . Applying Theorem 4.1 with n = 1 to the contin uous function F on the compact set K , there exist N ∈ N and constants a i , b i , c i ∈ R A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 11 suc h that      F ( ζ ) − N X i =1 c i σ ( a i ζ + b i )      < δ, for all ζ ∈ K . Define ˆ V ( ζ ) = ϕ N X i =1 c i σ ( a i ζ + b i ) ! . By the choice of δ ,    V ( ζ ) − ˆ V ( ζ )    < ε for all ζ ∈ K, whic h completes the pro of. □ Using this lemma, w e can no w establish that the tra v eling w a ve solution can b e represen ted by the proposed solver. Theorem 4.4 (Universal appro ximation for trav eling w av e solutions) . L et Ω ⊂ R b e a c omp act set and let ˆ v : Ω × [0 , T ] → R denote the solver define d in (11) . Then for any ε > 0 , ther e exist an inte ger N , c onstants a i , b i , c i ∈ R for i = 1 , . . . , N and a pr e dicte d wave sp e e d ω ∈ R , such that for al l ξ ∈ Ω , τ ∈ [0 , T ] , | v ( ξ , τ ) − ˆ v ( ξ , τ ) | < ε, wher e v : Ω × [0 , T ] → ( v − , v + ) is the tr aveling wave solution of the sc ale d r e action- diffusion e quation v τ = v ξξ + v p (1 − v q )( v − a ) r , p, q > 0 , r ∈ { 0 , 1 } , a ∈ (0 , 1) , with wave sp e e d c . Pr o of. Define K = { ξ − cτ | ξ ∈ Ω , τ ∈ [0 , T ] } . Then K ⊂ R is a compact. Let V denote the trav eling w av e profile defined b y V ( ζ ) = v ( ξ , τ ) , ζ = ξ − cτ . Since v is a tra veling wa v e solution, V is well defined on K and satisfies V ( ζ ) ∈ ( v − , v + ) for all ζ ∈ K . The function ϕ in (10) is con tin uous, surjectiv e and strictly monotone. Thus, b y Lemma 4.3 with ( α, β ) = ( v − , v + ), there exists a function ˆ V ( ζ ) = ϕ N X i =1 c i σ ( a i ζ + b i ) ! suc h that for all ζ ∈ K ,    V ( ζ ) − ˆ V ( ζ )    < ε. Define ˆ v ( ξ , τ ) = ˆ V ( ξ − cτ ) and choose ω = c . Then for all ξ ∈ Ω and τ ∈ [0 , T ], | v ( ξ , τ ) − ˆ v ( ξ , τ ) | < ε. □ Theorem 4.4 sho ws that the prop osed architecture can approximate tra veling w av e solutions of the scaled reaction-diffusion equation with arbitrary accuracy , pro vided that the predicted wa v e sp eed ω coincides with the exact wa v e sp eed c . 12 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG 5. Numerical resul ts for scaled TW-PINN W e present one- and tw o-dimensional numerical re sults to demonstrate the p o- ten tial of the prop osed scaling PINN framew ork for trav eling wa v e solutions, re- ferred to as scaled TW-PINN. The diffusion co efficient is fixed as D = 1 in this section. T o assess the accuracy of the scaled TW-PINN, w e make use of L 2 and L ∞ error norms: L 2 = v u u t 1 N N X i =1  u ( i ) − ˆ u ( i )  2 , L ∞ = max 1 ⩽ i ⩽ N    u ( i ) − ˆ u ( i )    , where u ( i ) denotes the exact solution and ˆ u ( i ) denotes the PINN appro ximation at the i -th collo cation p oint of N samples. 5.1. One-dimensional numerical results. Since the diffusion co efficien t D is fixed, the spatial and temp oral domains for numerical exp eriments dep end only on the reaction coefficient ρ . T able 5 summarizes the domains for each reaction- diffusion equation and eac h v alue of ρ , with the final time chosen so that the tra veling wa v e front remains within the spatial domain. T able 5. Spatial and temp oral domains for eac h reaction- diffusion equation and each v alue of ρ . v ariable ρ 1 10 2 10 4 10 6 Fisher x, y [ − 5 , 25] [ − 1 , 5] [ − 1 , 5] [ − 1 , 5] t [0 , 10] [0 , 0 . 2] [0 , 0 . 02] [0 , 0 . 002] NWS x, y [ − 5 , 25] [ − 1 , 5] [ − 1 , 5] [ − 1 , 5] t [0 , 10] [0 , 0 . 2] [0 , 0 . 02] [0 , 0 . 002] Zeldo vich x, y [ − 5 , 25] [ − 1 , 5] [ − 1 , 5] [ − 1 , 5] t [0 , 30] [0 , 0 . 6] [0 , 0 . 06] [0 , 0 . 006] bistable x, y [ − 25 , 5] [ − 5 , 1] [ − 5 , 1] [ − 5 , 1] t [0 , 25] [0 , 0 . 5] [0 , 0 . 05] [0 , 0 . 005] F or each v alue of ρ , 500 collo cation p oints are uniformly sampled in b oth space and temporal time, giving a total of 250 , 000 collo cation p oin ts. The accuracy test uses ten ph ysically con vergen t PINN solvers, with differen t random seed initializa- tions, trained on the original and restricted domains, as describ ed in Section 3.3. The mean and standard deviation of the L 2 and L ∞ errors for Fisher’s, NWS, Zel- do vich and bistable equations at different v alues of ρ are listed in T ables 6 and 7, resp ectiv ely . In general, the error increases as ρ b ecomes larger. How ever, on the restricted domain, NWs and bistable equations achiev es the smallest L 2 error at ρ = 10 4 . The scaled TW-PINNs on the restricted domain consistently give more accurate solutions, which can b e attributed to their impro v ed estimates of the pre- dicted wa ve sp eed ω , as shown in Section 3.3. Notably , for eac h equation, the L 2 errors calculated from sc aled TW-PINNs on the restricted domain at ρ = 10 6 are smaller than those from scaled TW-PINNs on the original domain at ρ = 1. These observ ations indicate that improv ed estimation of ω leads to more reliable n umerical A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 13 solutions. Moreov er, at ρ = 10 6 , the errors by scaled TW-PINNs on the restricted domain are of comparable magnitude across all equations. In contrast, for scaled TW-PINNs on the original domain, the NWS equation exhibits significantly larger errors than the other equations. T able 6. L 2 error for one-dimensional Fisher’s, NWS ( q = 2), Zeldo vich and bistable ( a = 0 . 2) equations. ρ 1 10 2 10 4 10 6 Fisher original 5.21e-5 (2.41e-5) 6.60e-5 (3.46e-5) 1.76e-4 (1.18e-4) 5.55e-4 (3.75e-4) restricted 5.33e-6 (1.52e-6) 7.52e-6 (1.77e-6) 6.49e-6 (1.54e-6) 1.04e-5 (5.36e-6) NWS original 1.13e-4 (8.40e-5) 1.54e-4 (1.20e-4) 4.75e-4 (3.86e-4) 1.50e-3 (1.22e-3) restricted 6.02e-6 (2.37e-6) 6.14e-6 (2.21e-6) 5.67e-6 (2.05e-6) 1.12e-5 (5.89e-6) Zeldovic h original 6.55e-5 (2.44e-5) 7.69e-5 (3.71e-5) 2.02e-4 (1.39e-4) 6.37e-4 (4.45e-4) restricted 1.00e-5 (4.50e-6) 1.01e-5 (4.53e-6) 1.01e-5 (8.15e-6) 2.18e-5 (2.70e-5) bistable original 5.04e-5 (3.09e-5) 5.54e-5 (4.17e-5) 1.24e-4 (1.42e-4) 3.88e-4 (4.53e-4) restricted 6.77e-6 (2.62e-6) 7.41e-6 (2.37e-6) 6.62e-6 (1.97e-6) 1.15e-5 (5.54e-6) T able 7. L ∞ error for one-dimensional Fisher’s, NWS ( q = 2), Zeldo vich and bistable ( a = 0 . 2) equations. ρ 1 10 2 10 4 10 6 Fisher original 1.80e-4 (9.90e-5) 3.27e-4 (2.07d-4) 3.14e-3 (2.12e-3) 3.06e-2 (2.07e-2) restricted 1.82e-5 (5.70e-6) 1.91e-5 (5.13e-6) 5.02e-5 (3.60e-5) 4.47e-4 (3.60e-4) NWS original 5.72e-4 (3.74e-4) 1.14e-3 (7.62e-4) 1.15e-2 (7.72e-3) 1.12e-1 (7.35e-2) restricted 1.39e-5 (5.52e-6) 1.42e-5 (5.37e-6) 8.10e-5 (4.83e-5) 8.01e-4 (4.83e-4) Zeldovic h original 2.42e-4 (1.27e-4) 4.39e-4 (2.82e-4) 4.30e-3 (2.95e-3) 4.29e-2 (2.94e-2) restricted 2.65e-5 (1.32e-5) 2.79e-5 (1.67e-5) 1.33e-4 (1.83e-4) 1.31e-3 (1.83e-3) bistable original 1.58e-4 (1.28e-4) 2.64e-4 (2.67e-4) 2.39e-3 (2.79e-3) 2.37e-2 (2.77e-2) restricted 1.73e-5 (7.68e-6) 1.75e-5 (7.52e-6) 6.04e-5 (3.91e-5) 5.71e-4 (4.18e-4) As seen in Section 3.3, the scaled TW-PINN can accurately predict the w a v e sp eed for the NWS equation even when q ≥ 2. T able 8 calculates the L 2 and L ∞ errors for the NWS equations with q = 3 , 4 under differen t v alues of ρ . These results are computed using one PINN solver for eac h v alue of q trained on the restricted domain. The resulting errors are significantly smaller than those b y the central WENO sc heme [10], where the numerical solution exhibits a noticeable lag b ehind the exact solution for q ≥ 2. This comparison further indicates that the scaled TW-PINNs on the restricted domain capture the wa ve fron t more accurately . In Fig. 5, we present the scaled TW-PINN solution at the final time sp ecified in T able 5 for each equation with ρ = 10 6 , compared to the corresp onding exact solution. F or each equation, the solution is obtained from a representativ e scaled TW-PINN on the original domain. It is seen that our scaled TW-PINN accurately captures the sharp wa v e fron t. T o further illustrate the error b ehavior, w e plot the maxim um error ov er the spatial domain against time t in Fig. 6, whic h shows the temporal ev olution of this maxim um error on a logarithmic scale for the Fisher’s, NWS, Zeldovic h and bistable equations. Unlike classical numerical theory , where the error typically accumulates o ver time, the maxim um error in scaled TW-PINN solutions do es not necessarily increase monotonically . 14 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG T able 8. L 2 and L ∞ errors for NWS equation ( q = 3 , 4) using scaled TW-PINNs on the restricted domain. q norm ρ 1 10 2 10 4 10 6 3 L 2 6.39e-6 5.78e-6 4.39e-6 8.81e-6 L ∞ 1.23e-5 1.44e-5 6.15e-5 6.00e-4 4 L 2 1.36e-5 1.26e-5 8.48e-6 2.08e-5 L ∞ 3.67e-5 3.67e-5 1.69e-4 1.53e-3 Fisher’s NWS ( q = 2) Zeldovic h bistable ( a = 0 . 2) 0 2 4 x 0 0.2 0.4 0.6 0.8 1 u ( x , T ) Exact scaled TW -PINN 0 2 4 x 0 0.2 0.4 0.6 0.8 1 Exact scaled TW -PINN 0 2 4 x 0 0.2 0.4 0.6 0.8 1 Exact scaled TW -PINN 4 2 0 x 0.2 0.4 0.6 0.8 1 Exact scaled TW -PINN Figure 5. Solution profiles, from left to right, for Fisher’s equa- tion at T = 0 . 002, NWS equation ( q = 2) at T = 0 . 002, Zel- do vich equation at T = 0 . 006, and bistable equation ( a = 0 . 2) at T = 0 . 005. The dashed black line is the exact solution. 5.2. Tw o-dimensional numerical results. In t wo-dimensional problems, we ev al- uate the L 2 and L ∞ errors by v arying the reaction co efficient ρ , giv en tw o prop- agation direction v ectors n . The spatial and temporal domains for each reaction- diffusion equation and eac h v alue of ρ are chosen to b e the same as in one-dimensional problems, listed in T able 5. F or eac h v alue of ρ , 100 collo cation points are uniformly sampled in b oth the spatial and temporal directions, generating a total of 10 6 collo cation p oints. W e use the same PINN solv ers trained on the original and restricted domains as in Section 5.1. The L 2 and L ∞ errors, in terms of mean and standard deviation, for Fisher’s, NWS, Zeldovic h and bistable equations at ρ and n are computed in T ables 9 and 10, resp ectively . The propagation direction n is selected from { (1 , 1) , (1 , 3) } and normalized to unit length. Since the direction (3 , 1) is symmetric to (1 , 3), it pro duces equiv alent results and is omitted. The error gro ws generally with ρ , consisten t with the one-dimensional results, while showing little v ariation with resp ect to n . This is expected b ecause the same PINN solv er for stage tw o is used in b oth one- and t wo-dimensional problems, and the remaining difference comes only from n and the num b er of collo cation p oints. Figure 7 presents the absolute error b etw een the exact and numerical solutions at the final time for eac h equation with ρ = 10 4 , 10 6 and n = (1 , 1). W e use a represen tative scaled TW-PINN on the original domain. As shown in Figure 7, larger errors o ccur near the wa v e fron t, but the error remains small, which means that the scaled TW-PINN captures the tra veling wa ve front accurately . W e also compute the maximum spatial error ov er t , but omit the resulting curves b ecause they are similar to those in Fig. 6 for the one-dimensional case. A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 15 Fisher’s NWS ( q = 2) Zeldovic h bistable ( a = 0 . 2) 0 2 4 6 8 10 t 4e-5 1e-4 2e-4 l o g 1 0 | u ( , t ) u ( , t ) | 0 2 4 6 8 10 t 9e-5 1.2e-4 1.7e-4 0 10 20 30 t 6e-5 4e-5 2.5e-5 0 5 10 15 20 25 t 5.5e-5 8e-5 1e-4 1.3e-4 0 0.05 0.1 0.15 0.2 t 6e-5 1e-4 2e-4 3e-4 l o g 1 0 | u ( , t ) u ( , t ) | 0 0.05 0.1 0.15 0.2 t 9e-5 2e-4 3.3e-4 0 0.2 0.4 0.6 t 1.46e-4 1.45e-4 0 0.1 0.2 0.3 0.4 0.5 t 7.6e-5 1.3e-4 2.3e-4 0 0.005 0.01 0.015 0.02 t 7e-5 3e-4 1e-3 3e-3 l o g 1 0 | u ( , t ) u ( , t ) | 0 0.005 0.01 0.015 0.02 t 8e-5 3e-4 1e-3 3e-3 0 0.02 0.04 0.06 t 3.4e-4 2.4e-4 1.4e-4 0 0.01 0.02 0.03 0.04 0.05 t 7.6e-5 4e-4 2e-3 0 0.0005 0.001 0.0015 0.002 t 3e-5 3e-4 3e-3 3e-2 l o g 1 0 | u ( , t ) u ( , t ) | 0 0.0005 0.001 0.0015 0.002 t 5e-5 1e-3 3e-2 0 0.002 0.004 0.006 t 3e-5 3e-4 3e-3 0 0.001 0.002 0.003 0.004 0.005 t 2.5e-5 1e-4 1e-3 2e-2 Figure 6. Ev olution of log-scale maximum errors ov er time, from left to right, for Fisher’s, NWS ( q = 2), Zeldovic h, and bistable ( a = 0 . 2) equations. The rows corresp ond to ρ = 1 , 10 2 , 10 4 , and 10 6 from top to b ottom. 5.3. Comparison with wa ve-PINN. W e compare the proposed scaled TW- PINN with the existing w a v e-PINN metho d in [24], whic h was designed sp ecifi- cally for the Fisher’s equation with the trav eling wa ve solution. The wa ve lay er for w av e-PINN defines the tra veling wa v e co ordinate as ˆ z = ω 1 √ ρ x + ω 2 ρ t + ω 3 . Then the reaction co efficient is incorp orated into the netw ork input, enabling ap- pro ximation for different ρ while D is fixed. Consequently the w a ve-PINN must b e retrained if D changes, which increases computational cost and limits the reusabil- it y of the trained netw ork. Moreov er, this co ordinate definition is applicable only to one spatial dimension because it directly uses the input v ariables x and t without a mechanism for extension to higher dimensions. In contrast, our wa v e lay er (12) op erates on scaled v ariables and is independent of spatial dimension. Then a single scaled TW-PINN can handle v arious physical co efficien ts and multiple spatial di- mensions. Figure 8 compares the the structure of the wa v e lay er in wa v e-PINN and 16 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG T able 9. L 2 error for tw o-dimensional Fisher’s, NWS ( q = 2), Zeldo vich and bistable ( a = 0 . 2) equations. n ρ 1 10 2 10 4 10 6 Fisher original (1,1) 5.07e-5 (2.63e-5) 6.80e-5 (3.82e-5) 1.91e-4 (1.28e-4) 5.35e-4 (3.61e-4) (1,3) 5.02e-5 (2.46e-5) 6.58e-5 (3.57e-5) 1.81e-4 (1.21e-4) 5.67e-4 (3.82e-4) restricted (1,1) 5.12e-6 (1.50e-6) 6.50e-6 (1.36e-6) 6.36e-6 (1.48e-6) 9.95e-6 (5.18e-6) (1,3) 5.14e-6 (1.50e-6) 6.94e-6 (1.53e-6) 6.36e-6 (1.47e-6) 1.04e-5 (5.55e-6) NWS original (1,1) 1.20e-4 (8.98e-5) 1.65e-4 (1.29e-4) 5.60e-4 (4.54e-4) 5.28e-3 (4.17e-3) (1,3) 1.16e-4 (8.60e-5) 1.58e-4 (1.23e-4) 4.87e-4 (3.95e-4) 1.53e-3 (1.25e-3) restricted (1,1) 6.29e-6 (2.37e-6) 6.36e-6 (2.19e-6) 6.19e-6 (2.31e-6) 3.70e-5 (2.15e-5) (1,3) 6.21e-6 (2.37e-6) 6.27e-6 (2.19e-6) 5.81e-6 (2.08e-6) 1.14e-5 (5.97e-6) Zeldovic h original (1,1) 7.04e-5 (2.48e-5) 8.08e-5 (3.83e-5) 2.33e-4 (1.59e-4) 2.24e-3 (1.54e-3) (1,3) 6.81e-5 (2.38e-5) 7.80e-5 (3.64e-5) 2.03e-4 (1.37e-4) 6.34e-4 (4.34e-4) restricted (1,1) 1.11e-5 (5.29e-6) 1.07e-5 (4.87e-6) 1.08e-5 (9.44e-6) 6.95e-5 (9.55e-5) (1,3) 1.08e-5 (4.92e-6) 1.04e-5 (4.65e-6) 1.01e-5 (8.08e-6) 2.14e-5 (2.64e-5) bistable original (1,1) 5.25e-5 (3.28e-5) 5.80e-5 (4.46e-5) 1.38e-4 (1.58e-4) 1.25e-3 (1.45e-3) (1,3) 5.14e-5 (3.16e-5) 5.63e-5 (4.26e-5) 1.27e-4 (1.46e-4) 3.96e-4 (4.62e-4) restricted (1,1) 7.39e-6 (3.15e-6) 7.90e-6 (2.82e-6) 7.05e-6 (2.19e-6) 3.13e-5 (2.07e-5) (1,3) 7.19e-6 (2.97e-6) 7.66e-6 (2.59e-6) 6.80e-6 (2.12e-6) 1.18e-5 (5.65e-6) T able 10. L ∞ error for tw o-dimensional Fisher’s, NWS ( q = 2), Zeldo vich and bistable ( a = 0 . 2) equations. n ρ 1 10 2 10 4 10 6 Fisher original (1,1) 1.80e-4 (9.90e-5) 3.26e-4 (2.07e-4) 3.10e-3 (2.09e-3) 2.45e-2 (1.66e-2) (1,3) 1.80e-4 (9.90e-5) 3.27e-4 (2.07e-4) 3.14e-3 (2.12e-3) 2.96e-2 (2.00e-2) restricted (1,1) 1.84e-5 (5.43e-6) 1.91e-5 (5.13e-6) 4.97e-5 (3.57e-5) 3.55e-4 (2.86e-4) (1,3) 1.84e-5 (5.49e-6) 1.91e-5 (5.13e-6) 4.98e-5 (3.57e-5) 4.31e-4 (3.49e-4) NWS original (1,1) 5.21e-4 (4.05e-4) 1.04e-3 (8.29e-4) 1.04e-2 (8.42e-3) 9.92e-2 (7.69e-2) (1,3) 5.22e-4 (4.05e-4) 1.04e-3 (8.29e-4) 1.02e-2 (8.25e-3) 9.40e-2 (7.73e-2) restricted (1,1) 1.36e-5 (5.14e-6) 1.40e-5 (5.00e-6) 7.05e-5 (4.10e-5) 7.01e-4 (4.11e-4) (1,3) 1.36e-5 (5.14e-6) 1.40e-5 (5.01e-6) 6.99e-5 (4.01e-5) 6.28e-4 (3.68e-4) Zeldovic h original (1,1) 2.42e-4 (1.27e-4) 4.38e-4 (2.82e-4) 4.30e-3 (2.95e-3) 4.28e-2 (2.92e-2) (1,3) 2.42e-4 (1.27e-4) 4.39e-4 (2.82e-4) 4.22e-3 (2.90e-3) 3.85e-2 (2.64e-2) restricted (1,1) 2.65e-5 (1.32e-5) 2.77e-5 (1.65e-5) 1.32e-4 (1.83e-4) 1.31e-3 (1.83e-3) (1,3) 2.65e-5 (1.32e-5) 2.79e-5 (1.67e-5) 1.30e-4 (1.79e-4) 1.17e-3 (1.64e-3) bistable original (1,1) 1.58e-4 (1.27e-4) 2.63e-4 (2.67e-4) 2.39e-3 (2.79e-3) 2.37e-2 (2.75e-2) (1,3) 1.58e-4 (1.28e-4) 2.64e-4 (2.67e-4) 2.36e-3 (2.76e-3) 2.15e-2 (2.50e-2) restricted (1,1) 1.73e-5 (7.68e-6) 1.75e-5 (7.52e-6) 6.00e-5 (3.89e-5) 5.73e-4 (4.18e-4) (1,3) 1.73e-5 (7.68e-6) 1.75e-5 (7.52e-6) 5.97e-5 (3.87e-5) 5.13e-4 (3.75e-4) scaled TW-PINN. In addition, wa v e-PINN and scaled TW-PINN differ in the num- b er of hidden lay ers: the former uses three hidden lay ers, while the latter uses only one. F or a fair comparison, wa ve-PINN is trained under a configuration similar to that of the proposed scaled TW-PINN while keeping its original arc hitecture. The net work contains three hidden la y ers with 20 neurons each and applies the logistic sigmoid activ ation function. In training, w e tak e 1024 samples from ρ ∈ (1 , 10 6 ) and the initial learning rate is 0 . 001. T able 11 presents the L 2 and L ∞ errors of w a ve-PINN and scaled TW-PINN on the original domain for the Fisher’s equation. As sho wn in T able 11, our scaled TW-PINN achiev es significantly higher accuracy for all v alues of ρ . The approxi- mate solutions for ρ = 10 6 at the initial time t = 0 and the final time T = 0 . 002, along with the p oint wise errors on a logarithmic scale, are plotted in Fig. 9. W e A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 17 Fisher’s NWS ( q = 2) Zeldovic h bistable ( a = 0 . 2) 0 2 4 x 0 2 4 y 0 2 4 x 0 2 4 y 0 2 4 x 0 2 4 y 4 2 0 x 4 2 0 y 0 2 4 x 0 2 4 y 0 2 4 x 0 2 4 y 0 2 4 x 0 2 4 y 4 2 0 x 4 2 0 y 1 0 5 1 0 4 1 0 3 1 0 5 1 0 4 1 0 3 1 0 5 1 0 4 1 0 3 1 0 2 1 0 5 1 0 4 1 0 3 1 0 2 1 0 5 1 0 4 1 0 3 1 0 5 1 0 4 1 0 3 1 0 5 1 0 4 1 0 3 1 0 2 1 0 5 1 0 4 1 0 3 1 0 2 Figure 7. Absolute error b etw een the exact and scaled TW-PINN solutions in the filled contour plot, from left to right, for Fisher’s equation at T = 0 . 002, NWS equation ( q = 2) at T = 0 . 002, Zeldo vich equation at T = 0 . 006, and bistable equation ( a = 0 . 2) at T = 0 . 005 with ρ = 10 4 (top) and ρ = 10 6 (b ottom). Figure 8. Comparison of the structure of the w a ve la yer in wa ve- PINN (left), scaled TW-PINN (middle) and scaled gTW-PINN (righ t). T able 11. L 2 and L ∞ errors of w av e-PINN and scaled TW-PINN for Fisher’s equation. norm ρ 1 10 2 10 4 10 6 wa ve-PINN L 2 2.23e-3 (1.67e-3) 3.11e-3 (2.35e-3) 9.72e-3 (7.32e-3) 2.37e-2 (1.44e-2) L ∞ 8.95e-3 (6.75e-3) 1.77e-2 (1.34e-2) 1.71e-1 (1.26e-1) 7.64e-1 (2.94e-1) scaled TW-PINN L 2 5.21e-5 (2.41e-5) 6.60e-5 (3.46e-5) 1.76e-4 (1.18e-4) 5.55e-4 (3.75e-4) L ∞ 1.80e-4 (9.90e-5) 3.27e-4 (2.07e-4) 3.14e-3 (2.12e-3) 3.06e-2 (2.07e-2) can see that b oth w a ve-PINN and scaled TW-PINN pro duce accurate solutions at the initial time t = 0. How ever, at the final time, w av e-PINN predicts the w av e front with an excessive sp eed, causing the fron t to mov e ahead of its correct lo cation, whereas the prop osed scaled TW-PINN pro vides a more accurate approx- imation. While wa v e-PINN ac hiev es higher accuracy near the equilibrium states, it exhibits substantially larger errors in the transition la y er. In contrast, scaled TW-PINN sho ws sligh tly larger errors near the equilibrium states but main tains stable accuracy across the transition lay er. These differences app ear to arise from the residual weigh ting scheme in w av e-PINN. By assigning smaller w eights to the 18 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG 1 0 1 2 3 4 5 x 0 0.2 0.4 0.6 0.8 1 u ( x , T = 0 ) 0.02 0.01 0.00 0.01 0.02 0 0.2 0.4 0.6 0.8 1 Exact wave-PINN scaled TW -PINN 1 0 1 2 3 4 5 x 0 0.2 0.4 0.6 0.8 1 u ( x , T = 0 . 0 0 2 ) 4.06 4.07 4.08 4.09 4.10 0 0.2 0.4 0.6 0.8 1 Exact wave-PINN scaled TW -PINN 1 0 1 2 3 4 5 x 1e-10 1e-8 1e-6 1e-4 1e-2 1e-0 l o g 1 0 | u ( , 0 ) u ( , 0 ) | wave-PINN scaled TW -PINN 1 0 1 2 3 4 5 x 1e-10 1e-8 1e-6 1e-4 1e-2 1e-0 l o g 1 0 | u ( , T ) u ( , T ) | wave-PINN scaled TW -PINN Figure 9. Solution profiles (top) and log-scale p oint wise errors (b ottom) for Fisher’s equation with ρ = 10 6 at the initial time t = 0 (left) and the final time T = 0 . 002 (right) appro ximated b y w av e-PINN (cyan) and scaled TW-PINN (red). The dashed blac k line is the exact solution. sharp region, w a ve-PINN tends to learn the equilibrium regions more accurately than the w av e front. In con trast, the scaling transformation in scaled TW-PINN smo oths the sharp transition, allo wing the PINN solv er in stage t wo to ac hieve more balanced learning across the spatial domain. These results demonstrate that scaled TW-PINN more reliably predicts the trav eling wa v e b ehavior than wa ve-PINN, particularly in terms of wa v e front position and wa v e sp eed. 6. Extension to Fisher ’s equa tion with general initial conditions Previously , w e only consider the specific initial condition for which the reaction- diffusion equation admits an exact tra v eling w av e solution in closed form with a sp ecial wa ve speed. W e now turn to more general initial conditions. W e fo cus on the one-dimensional scaled Fisher’s equation, v τ = v ξξ + v (1 − v ) , sub ject to the initial condition v ( ξ , 0) = v 0 ( ξ ). Notice that the Fisher’s equation is c hosen as its long time b ehavior of the tra v eling w av e solution under the general initial condition is extensively studied in the literature [2, 19]. The same framew ork also extends to the other equations in this paper. Supp ose that the initial condition satisfies (13) v 0 ( ξ ) ∼ exp( − λξ ) as ξ → ∞ , with λ > 0. By [19], the solution ev olves to a tra v eling w a v e whose asymptotic w av e sp eed is c ( λ ) =  λ + 1 λ , 0 < λ < 1 , 2 , λ ≥ 1 . F or 0 < λ < 1, the asymptotic behavior is not fully c haracterized. In the case λ ≥ 1 corresp onding to the minimum wa ve sp eed c = 2, the long time behavior of the A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 19 solution [2] is describ ed by (14) v ( ξ , τ ) → V  ξ − 2 τ + 3 2 ln τ + O (1)  as τ → ∞ , with O (1) a b ounded spatial shift. Th us, the wa ve sp eed is not constan t. Accord- ingly , a generalized tra veling wa v e co ordinate is introduced as ζ = ξ − d ( τ ) , where d ( τ ) represen ts a time-dependent shift and the wa v e speed is given b y d ′ ( τ ). F or the constant wa ve sp eed, one sets d ( τ ) = cτ , which gives d ′ ( τ ) = c . Based on this formulation, the w a v e lay er (9) is generalized to ˆ ζ = ξ − ˆ d ( τ ) , as shown on the right of Fig. 8. F ollowing (14) giv es the definition of the predicted w av e shift (15) ˆ d ( τ ; λ ) = c ( λ ) τ − 3 2 ln( τ + 1) + w λ, where λ corresp onds to the exp onential deca y of the initial condition (13), and c ( λ ) denotes the asymptotic wa ve speed (14). The term ln( τ + 1) a v oids the singularity at τ = 0 while preserving the same asymptotic b ehavior for large τ , and the final term wλ in tro duces a trainable correction that accounts for the b ounded spatial shift O (1). Although the asymptotic b ehavior is known only for λ ≥ 1, w e adopt the form (14) here as a heuristic appro ximation and inv estigate whether it remains informativ e for describing the wa v e shift under 0 < λ < 1. W e consider three initial conditions: (i) a step function v 0 ( ξ ) =  1 , ξ < 0 , 0 , ξ ≥ 0 , (ii) a logistic-type function of the form (16) v 0 ( ξ ) = 1  1 + exp( λ 2 ξ )  2 , with λ = 2, and (iii) the logistic-t yp e function (16) with λ = 1 2 . The tw o v alues of λ corresp ond to λ ≥ 1 and 0 < λ < 1, resp ectively . The initial condition of the step function has a rapidly deca ying leading edge and b ehav es similarly to initial conditions with λ ≥ 1. W e take the PINN architecture as in Section 3.1 with the wa v e lay er replaced by (15). The same loss function in Section 3.2 is emplo yed, and Diric hlet b oundary conditions are imposed according to (7). Since a relatively small training domain with the fo cus on the wa ve front promotes rapid physical conv ergence, we choose the training ( ξ , τ )-domain as [ − 300 , 900] × [0 , 300], and pic k N ICBC = 1024 and N r = 1024 collo cation p oints for L ICBC and L r , resp ectively . The generalized PINN solver is trained for 30 , 000 ep o c hs. A t early times, the solution deforms from the initial condition. Sp ecifically , p oin ts along the wa v e front propagate at differen t speeds dep ending on the initial condition, resulting in a deformation of the w av e front shap e. F or initial conditions (i) and (ii), matching λ ≥ 1, the wa ve front propagates more rapidly near its leading edge, whereas in the initial condition (iii), corresp onding to 0 < λ < 1, the front mov es faster near its trailing edge. Since the PINN solver is designed for tra v eling wa ves, it is not easy to capture such deformation. T o address this, the initial and b oundary loss, as well as the residual 20 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG loss, is applied during the first 0 . 3 ep o chs of training, after whic h only the residual loss is used. All other training settings are the same as in Section 3.3. The scaling PINN framework for general initial conditions, termed scaled gTW- PINN, is used in the follo wing n umerical experiments. W e consider the Fisher’s equation with ρ = 10 2 and ρ = 10 4 , and approximate the corresponding scaled gTW-PINN solutions at final times T = 0 . 3 and T = 0 . 03. Since the exact solu- tion is not av ailable for Fisher’s equation with these general initial conditions, the reference solution is computed by the cen tral WENO scheme [23, 10] with 2000 uniform spatial cells on the domain [ − 3 , 9]. Figure (10) shows the n umerical results for ρ = 10 2 . W e see that scaled gTW-PINN captures the wa v e front well for the initial conditions (i) and (ii) with λ ≥ 1, v alidating the asymptotic b eha vior (14). F or the initial condition (iii), the solution with scaled gTW-PINN trav els b ehind the reference wa ve front, and the phase error is noticeable. This discrepancy may b e caused by the fact that the predicted wa ve shift (15) from the asymptotic form (14), is not well suited to the case 0 < λ < 1. In particular, the logarithmic phase correction and the accompan ying O (1) spatial shift ma y introduce an inaccurate w av e shift. W e plot the numerical solution for ρ = 10 4 b y scaled gTW-PINN to- gether with the reference solution in Fig. (10). It is observed that the wa v e fron t appro ximated b y scaled gTW-PINN agrees with the reference solution for all three initial conditions. 2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 u(x,T=0.3) R efer ence scaled gTW -PINN 2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 R efer ence scaled gTW -PINN 2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 R efer ence scaled gTW -PINN Figure 10. Solution profiles for Fisher’s equation ( ρ = 10 2 ) sub- ject to the initial condition (i) the step function (left), (ii) (16) with λ = 2 (middle) and (iii) (16) with λ = 1 2 (righ t) at the final time T = 0 . 3 appro ximated by scaled gTW-PINN (red). The dashed blac k line is the reference solution. 2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 u(x,T=0.03) R efer ence scaled gTW -PINN 2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 R efer ence scaled gTW -PINN 2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 R efer ence scaled gTW -PINN Figure 11. Solution profiles for Fisher’s equation ( ρ = 10 4 ) sub- ject to the initial condition (i) the step function (left), (ii) (16) with λ = 2 (middle) and (iii) (16) with λ = 1 2 (righ t) at the final time T = 0 . 03 approximated b y scaled gTW-PINN (red). The dashed blac k line is the reference solution. A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 21 7. Conclusion In this pap er, w e dev elop a scaling PINN framework to solv e the reaction–diffusion equation with the trav eling wa v e solution. F or the sharp wa v e fron t from the large reaction co efficient, we apply a scaling transformation that normalizes the reaction and diffusion co efficients, employ a PINN solv er for the resulting scaled equation, and recov er the solution of the original equation via an inv erse transformation. The wa ve lay er in the PINN solver is designed to preserve the trav eling wa ve form, even in higher spatial dimensions. W e also show that the prop osed PINN solv er has a univ ersal approximation prop erty for trav eling w a v e solutions. One- and tw o-dimensional numerical experiments demonstrate that the prop osed scaled TW-PINN accurately captures the tra v eling wa ve solution, and the comparison to the existing w av e-PINN method show that it ac hieves higher accuracy and smaller errors near the wa ve front. F urthermore, the framework extends to general initial conditions, suggesting its broad applicability . F uture w ork will inv estigate the ap- plication of PINNs with scaling and inv erse transformations to reaction–diffusion systems exhibiting more complex phenomena, suc h as spiral and scroll w av es. A cknowledgments Jiaxi Gu is supported b y POSTECH Basic Science Research Institute F und, whose NRF grant num b er is RS-2021-NR060139. Jae-Hun Jung is supp orted b y Na- tional Research F oundation of Korea (NRF) under the grant num b er 2021R1A2C3009648, POSTECH Basic Science Researc h Institute under the NRF gran t num ber 2021R1A6A1A10042944, and partially NRF gran t funded by the Korea gov ernment (MSIT) (RS-2023-00219980). References [1] Donald G. Aronson and Hans F. W einberger, Nonline ar diffusion in population genetics, c om- bustion, and nerve pr opagation , Partial Differen tial Equations and Related T opics. Lecture Notes in Mathematics (J.A. Goldstein, ed.), Springer, Berlin, 1975, pp. 5–49. [2] Maury Bramson, Conver genc e of solutions of the kolmogor ov e quation to tr avel ling waves , American Mathematical Society , 1983. [3] Sung W o ong Cho, Hyung Ju Hwang, and Hwijae Son, T r aveling wave solutions of partial differ ential e quations via neural networks , J. Sci. Comput. 89 (2021), no. 1, 21. [4] G. Cybenko, Appr oximation by sup erp ositions of a sigmoidal function , Math. Control Signals Syst. 2 (1989), no. 4, 303–314. [5] Gerhard Ertl, Oscil latory kinetics and spatio-temp or al self-or ganization in re actions at solid surfac es , Science 254 (1991), no. 5039, 1750–1755. [6] Ric hard J. Field, Endre Koros, and Richard M. Noyes, Oscillations in chemic al systems. II. thor ough analysis of temp or al oscil lation in the br omate-c erium-malonic acid system , J. Am. Chem. Soc. 94 (1972), no. 25, 8649–8664. [7] P . C. Fife and J. B. McLeo d, The appr o ach of solutions of nonline ar diffusion e quations to tr avel ling fr ont solutions , Arc h. Rational Mec h. Anal. 65 (1977), 335–361. [8] R.A. Fisher, The wave of advanc e of advantage ous genes , Ann. Eugen. 7 (1937), no. 4, 355–369. [9] Ric hard FitzHugh, Impulses and physiolo gic al states in the or etic al mo dels of nerve membr ane , Biophys. J. 1 (1961), no. 6, 445–466. [10] Jiaxi Gu, Daniel Olmos-Liceaga, and Jae-Hun Jung, A numeric al study of WENO appr oxi- mations to sharp pr op agating fr onts for r e action-diffusion systems , Appl. Numer. Math. 211 (2025), 1–16. [11] A. L. Ho dgkin and A. F. Huxley , A quantitative description of membr ane curr ent and its applic ation to conduction and excitation in nerve , J. Ph ysiol. 117 (1952), no. 4, 500–544. 22 SEUNGW AN HAN, KW ANGHYUK P ARK, JIAXI GU, AND JAE-HUN JUNG [12] S. Jakubith, H. H. Roterm und, W. Engel, A. v on Oertzen, and G. Ertl, Sp atiotemp or al c on- c entration p atterns in a surfac e r e action: Pr op agating and standing waves, r otating spir als, and turbulence , Ph ys. Rev. Lett. 65 (1990), 3013–3016. [13] S.A. Kauffman, Metab olic stability and epigenesis in r andomly c onstructe d genetic nets , J. Theor. Biol. 22 (1969), no. 3, 437–467. [14] James P . Keener, A ge ometric al the ory for spir al waves in excitable me dia , SIAM J. Appl. Math. 46 (1986), no. 6, 1039–1956. [15] Seungc han Ko and Sanghy eon Park, VS-PINN: A fast and efficient tr aining of physics- informe d neural networks using variable-sc aling metho ds for solving PDEs with stiff b ehavior , J. Comput. Ph ys. 529 (2025), 113860. [16] Aditi S. Krishnapriy an, Amir Gholami, Shandian Zhe, Rob ert M. Kirb y , and Michael W. Ma- honey , Char acterizing possible failur e mo des in physics-informe d neur al networks , Advances in Neural Information Pro cessing Systems 34: Ann ual Conference on Neural Information Processing Systems 2021 (NeurIPS 2021), 2021, pp. 26548–26560. [17] Li Liu, Shengping Liu, Hui Xie, F ansheng Xiong, T engchao Y u, Meng juan Xiao, Lufeng Liu, and Heng Y ong, Disc ontinuity c omputing using physics-informe d neural networks , J. Sci. Comput. 98 (2024), 22. [18] Levi D. McClenny and Ulisses M. Braga-Neto, Self-adaptive physics-informed neur al net- works , J. Comput. Ph ys. 474 (2023), 111722. [19] J. D. Murra y , Mathematical biolo gy , Springer New Y ork, NY, 2002. [20] , Mathematic al biolo gy ii , Springer New Y ork, NY, 2003. [21] Q. Ouy ang and H. Swinney , T ransition from a uniform state to hexagonal and stripe d T uring p atterns , Nature (1991), no. 352, 610–612. [22] M. Raissi, P . P erdik aris, and G. E. Karniadakis, Physics-informe d neur al networks: A de ep le arning framework for solving forwar d and inverse pr oblems involving nonlinear p artial dif- fer ential e quations , J. Comput. Phys. 378 (2019), 686–707. [23] S. Rathan and J. Gu, A sixth-order c entr al WENO scheme for nonline ar degener ate p ar abolic e quations , Comp. Appl. Math. 42 (2023), 182. [24] F ranz M. Rohrhofer, Stefan P osch, Clemens G¨ oßnitzer, and Bernhard C. Geiger, Appr oximat- ing families of sharp solutions to Fisher’s e quation with physics-informe d neur al networks , Comput. Phys. Commun. 307 (2025), 109422. [25] J. G. Sk ellam, R andom disp ersal in theor etic al p opulations , Biometrik a 38 (1951), 196–218. [26] Alan M. T uring, The chemical b asis of morpho genesis , Phil. T rans. R. So c. Lond. B 237 (1952), no. 641, 37–72. [27] Wim v an Saarlo os, F r ont pr op agation into unstable states: Mar ginal stability as a dynamical me chanism for velocity sele ction , Phys. Rev. A 37 (1988), 211–229. [28] Sifan W ang, Y ujun T eng, and Paris P erdik aris, Understanding and mitigating gr adient flow p athologies in physics-informe d neur al networks , SIAM J. Sci. Comput. 43 (2021), no. 5, A3055–A3081. [29] Arth ur T. Winfree, Spir al waves of chemic al activity , Science 175 (1972), no. 4022, 634–636. [30] , Scr ol l-shap e d waves of chemical activity in thr e e dimensions , Science 181 (1973), no. 4103, 937–939. [31] Jac k Xin, F r ont pr opagation in hetero gene ous me dia , SIAM Rev. 42 (2000), no. 2, 161–230. A SCALED TW-PINN FOR TRA VELING W A VE SOLUTIONS 23 Dep ar tment of Ma thema tics & POSTECH MINDS (Ma thema tical Institute f or Da t a Science), Pohang University of Science and Technology, Pohang 37673, K orea Email address : han97@postech.ac.kr Gradua te School of Ar tificial Intelligence & POSTECH MINDS (Ma thema tical Institute for Da t a Science), Pohang University of Science and Technology, Pohang 37673, Korea Email address : pkh0219@postech.ac.kr Dep ar tment of Ma thema tics & POSTECH MINDS (Ma thema tical Institute f or Da t a Science), Pohang University of Science and Technology, Pohang 37673, K orea Email address : jiaxigu@postech.ac.kr Dep ar tment of Ma thema tics & POSTECH MINDS (Ma thema tical Institute f or Da t a Science), Pohang University of Science and Technology, Pohang 37673, K orea Email address : jung153@postech.ac.kr