Deepmechanics

Physics-informed deep learning models have emerged as powerful tools for learning dynamical systems. These models directly encode physical principles into network architectures. However, systematic benchmarking of these approaches across diverse physical phenomena remains limited, particularly in conservative and dissipative systems. In addition, benchmarking that has been done thus far does not integrate out full trajectories to check stability. In this work, we benchmark three prominent physics-informed architectures such as Hamiltonian Neural Networks (HNN), Lagrangian Neural Networks (LNN), and Symplectic Recurrent Neural Networks (SRNN) using the DeepChem framework, an open-source scientific machine learning library. We evaluate these models on six dynamical systems spanning classical conservative mechanics (mass-spring system, simple pendulum, double pendulum, and three-body problem, spring-pendulum) and non-conservative systems with contact (bouncing ball). We evaluate models by computing error on predicted trajectories and evaluate error both quantitatively and qualitatively. We find that all benchmarked models struggle to maintain stability for chaotic or nonconservative systems. Our results suggest that more research is needed for physics-informed deep learning models to learn robust models of classical mechanical systems.

💡 Research Summary

The paper “Deepmechanics” presents a systematic benchmark of three prominent physics‑informed neural network (PINN) architectures—Hamiltonian Neural Networks (HNN), Lagrangian Neural Networks (LNN), and Symplectic Recurrent Neural Networks (SRNN)—implemented within the open‑source DeepChem library. The authors argue that, despite growing interest in PINNs for learning dynamical systems, existing evaluations are fragmented, often limited to conservative systems, and rarely assess long‑term trajectory stability. To address these gaps, the authors extend DeepChem with high‑level abstractions (NumpyDataset, TorchModel) and integrate the three models into a unified training and evaluation pipeline.

Six dynamical systems are selected to span a broad spectrum of physical behavior: (1) a simple mass‑spring oscillator, (2) a simple pendulum, (3) a spring‑pendulum (coupled radial‑angular motion), (4) a double pendulum, (5) the three‑body problem, and (6) a bouncing ball that involves contact and energy loss. The first two are classic conservative systems with analytically known Hamiltonians/Lagrangians; the next three are chaotic or highly nonlinear conservative systems; the last is a non‑conservative system with discontinuous dynamics. For each system, synthetic trajectories are generated using high‑accuracy SciPy ODE solvers (solve_ivp with RK45, odeint) or a fixed‑step leapfrog integrator, and then split 80/20 into training and test sets.

Model specifics: HNN learns a scalar Hamiltonian H(q,p) and obtains time derivatives via automatic differentiation; LNN learns a scalar Lagrangian L(q, ẋ) and derives accelerations through the Euler‑Lagrange equations; SRNN embeds a discrete symplectic leapfrog step inside the network, directly predicting the next state while preserving the symplectic form. All networks share a four‑layer fully‑connected architecture with 128–256 hidden units, ReLU activations, and are trained on a single GPU for 100–1000 epochs using Adam (learning rate 1e‑3) and batch sizes of 32–64.

Evaluation metrics include mean squared error (MSE), mean absolute error (MAE), root‑mean‑square error (RMSE), standard deviation (STD), and variance (VAR) computed between predicted and ground‑truth trajectories, as well as qualitative phase‑space plots and energy‑over‑time curves.

Key findings:

• Conservative simple systems (mass‑spring, simple pendulum). HNN achieves the lowest errors on the mass‑spring task (MSE ≈ 0.012, RMSE ≈ 0.097) and competitive performance on the pendulum, while SRNN is slightly worse but still stable over long horizons. LNN shows higher numerical errors but preserves the overall shape of trajectories, indicating that gradient‑based reconstruction of dynamics can be less precise yet qualitatively acceptable.

• Chaotic systems (spring‑pendulum, double pendulum, three‑body). All three models struggle to maintain long‑term fidelity. For the spring‑pendulum, LNN attains the best quantitative scores (MSE ≈ 0.0003), HNN is acceptable, whereas SRNN’s error explodes (MSE ≈ 20). The authors attribute SRNN’s failure to its reliance on a fixed small time step and insufficient capacity to capture the strongly coupled nonlinear dynamics. Similar degradation is observed for the double pendulum and three‑body problem (details omitted but described as “rapid energy drift and trajectory divergence”).

• Non‑conservative contact system (bouncing ball). None of the models incorporate explicit dissipation or contact laws; consequently, predicted trajectories either conserve energy unrealistically after impact or produce unstable oscillations. The paper highlights this as a major limitation of current PINN formulations when applied to systems with discontinuities or friction.

Overall, the benchmark demonstrates that encoding physical priors (energy conservation, symplectic structure) improves short‑term prediction for smooth, low‑dimensional conservative dynamics, but does not guarantee stability for chaotic or dissipative systems. The authors conclude that future work must (i) integrate differentiable contact and damping models, (ii) explore hybrid physics‑data approaches that combine learned components with known non‑conservative terms, (iii) investigate adaptive or higher‑order symplectic integrators within the network, and (iv) expand the benchmark to include richer datasets and real‑world measurements.

By providing a reproducible DeepChem‑based framework, “Deepmechanics” offers the community a standardized testbed for comparing existing and forthcoming physics‑informed architectures, thereby accelerating progress toward robust, generalizable models of classical mechanics.