Known physics residual constraints improve nonlinear neural PDE models
When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics
Machine LearningArtificial Intelligence
Summary
This paper looks at how adding physics knowledge to neural network models that solve partial differential equations (PDEs) can help. The authors found that including known equation residuals as constraints improves model performance, especially for nonlinear dynamics like Burgers and KdV equations. However, this approach doesn’t help, and can even hurt, when modeling linear equations or when the data grid is too coarse. Other methods like transfer learning or in-context conditioning didn't outperform building models from scratch in their tests.
What this means in practice
- •For computational physicists: Design neural PDE models that use known physics residuals to improve predictions for nonlinear physical systems.
- •For engineering simulation teams: Build computational models with embedded physics constraints to enhance accuracy in nonlinear fluid and material dynamics simulations.
Authors
Zahra Farazpay, Aniruddha Bora
Abstract
Neural PDE surrogates increasingly incorporate structural priors, yet it is often unclear whether their gains arise from physics-specific information or simply from regularization and training choices. We evaluate several such priors under a common protocol against a matched from-scratch neural operator baseline. Our central result is that a known-equation residual consistently outperforms the best generic regularizer at equal tuning budget. At fixed capacity this benefit appears across linear and nonlinear PDEs, but a capacity sweep reveals a sharp distinction: the advantage persists and grows for Burgers, KdV, and Allen-Cahn, while collapsing toward or below parity for linear heat and advection-diffusion. Thus, the durable value of the residual is specific to nonlinear operators. We further falsify a pre-registered hypothesis that the benefit is activated only by data sparsity: the residual remains advantageous even under full supervision. Its usefulness does, however, have a clear boundary. Under grid under-resolution, nonlinear coarse fields no longer satisfy the naive governing-equation residual, and enforcing it becomes actively harmful. In contrast, cross-family pretraining and in-context conditioning fail to outperform the strong from-scratch baseline in the regime studied. Together, these results identify when known physics provides non-redundant information to neural PDE models, when it does not, and when enforcing it introduces bias.