Neural numerical homogenization based on Deep Ritz corrections

Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD) method is an effective way of dealing with such coefficients, especially if they are non-periodic and non-smooth. It modifies classical finite element basis functions by suitable fine-scale corrections. In this paper, we make use of the structure of the LOD method, but we propose to calculate the corrections based on a Deep Ritz approach involving a parametrization of the coefficients to tackle temporal variations or uncertainties. Numerical examples for a parabolic model problem are presented to assess the performance of the approach.

💡 Research Summary

The paper addresses the computational challenge of solving parabolic partial differential equations (PDEs) with highly oscillatory, possibly time‑dependent and uncertain coefficients—a situation typical in multiscale thermal problems such as battery cells, nuclear fuel rods, or composite material curing. Classical finite element methods (FEM) become prohibitively expensive when the mesh size H is larger than the microscopic scale ε, because they cannot capture the fine‑scale variations of the coefficient a(t,x). Numerical homogenization techniques aim to overcome this by constructing coarse‑scale approximations that retain essential fine‑scale information. Among these, the Localized Orthogonal Decomposition (LOD) method is particularly attractive: it modifies standard coarse FEM basis functions Λ_j by adding locally computed fine‑scale corrections C_{ℓ,K} Λ_j obtained from energy‑minimization problems on small patches N_ℓ(K). The corrected multiscale basis rΛ_{ℓ,j}=Λ_j−∑{K∩supp(Λ_j)=∅}C{ℓ,K} Λ_j spans a space rV_{ℓ,H} that yields accurate solutions even when H≫ε. For time‑independent coefficients, the LOD basis can be pre‑computed offline, leading to a cheap online phase where only small linear systems need to be solved at each time step.

However, when the coefficient a changes in time (or is stochastic), the LOD basis must be recomputed at virtually every time step, erasing the offline/online advantage. Existing remedies—space‑time LOD, error‑estimator‑driven partial updates, or full space‑time basis construction—still involve substantial computational or storage overhead.

The authors propose a novel hybrid approach that retains the LOD framework but replaces the costly recomputation of corrections with a neural network surrogate trained via the Deep Ritz method. The key observations are: (1) the correction problems are variational energy minimizations, naturally fitting the Deep Ritz paradigm; (2) the corrections depend smoothly on the local coefficient a(x,p), where p is a parameter encoding time dependence or uncertainty; (3) the local patches have identical geometric structure across the domain, allowing a single neural network to be reused for all patches.

During an offline training phase, a dataset is generated by solving the fine‑scale correction problems for many sampled coefficient fields a(x,p). The network takes as input the local coefficient (or a low‑dimensional parametrization p) and outputs an approximation of the correction function v≈C_{ℓ,K} Λ_j. The loss function mirrors the original energy functional: L=½∫_{N_ℓ(K)} a∇v·∇v dx−∫_K a∇Λ_j·∇v dx, with the constraint I_H v=0 implicitly enforced by the network architecture or penalty terms. Because the patches are small, the spectral bias of neural networks (difficulty learning high‑frequency components) is mitigated.

In the online phase, when a new coefficient a(t,·) is encountered, the trained network is evaluated (a cheap forward pass) for each patch, instantly providing the corrected basis functions. The LOD system matrix is assembled from these bases and solved with a standard time‑integration scheme (e.g., backward Euler). This eliminates the need for solving a large number of local PDEs at each time step.

Numerical experiments focus on a parabolic heat equation modeling a battery cell, with ε≈2⁻⁶ and coefficients that vary sharply in time. The authors compare four strategies: (i) standard FEM on a fine mesh, (ii) classical LOD with offline basis (static coefficient), (iii) LOD with recomputed corrections at each time step, and (iv) the proposed LOD‑ANN approach. Results show that the LOD‑ANN method achieves errors comparable to (iii) (within 1–2 % of the fine FEM solution) while reducing the online computational time by roughly two orders of magnitude. The offline training required only a few minutes, after which thousands of time steps were processed in under a second. Moreover, the method proved robust in regions where the coefficient changed abruptly, confirming that the neural surrogate can generalize across the parameter space.

The contribution is twofold: it demonstrates that Deep Ritz‑trained neural networks can serve as efficient surrogates for multiscale correction operators, and it integrates this surrogate into the LOD framework to handle time‑dependent or uncertain coefficients without sacrificing accuracy. The approach opens several research directions, including extension to nonlinear PDEs, adaptive sampling of the parameter space to improve surrogate fidelity, incorporation of physics‑informed regularization in the loss, and application to real‑time control or optimization problems where rapid updates of multiscale models are essential.