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.
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Cite this article
APA 7
Farazpay, Z., & Bora, A. (2026). When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics. https://omanscience.com/en/articles/when-known-physics-helps-neural-pde-models-residual-constraints-out-regularize-generic-priors-for-nonlinear-dynamics
MLA 9
Farazpay, Zahra, and Aniruddha Bora. "When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics." https://omanscience.com/en/articles/when-known-physics-helps-neural-pde-models-residual-constraints-out-regularize-generic-priors-for-nonlinear-dynamics.
Chicago (author–date)
Farazpay, Zahra, and Aniruddha Bora. 2026. "When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics." https://omanscience.com/en/articles/when-known-physics-helps-neural-pde-models-residual-constraints-out-regularize-generic-priors-for-nonlinear-dynamics.
Harvard
Farazpay, Z. and Bora, A. (2026) 'When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics', Available at: https://omanscience.com/en/articles/when-known-physics-helps-neural-pde-models-residual-constraints-out-regularize-generic-priors-for-nonlinear-dynamics.
Vancouver
Farazpay Z, Bora A. When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics. https://omanscience.com/en/articles/when-known-physics-helps-neural-pde-models-residual-constraints-out-regularize-generic-priors-for-nonlinear-dynamics
IEEE
Z. Farazpay, and A. Bora, "When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics," https://omanscience.com/en/articles/when-known-physics-helps-neural-pde-models-residual-constraints-out-regularize-generic-priors-for-nonlinear-dynamics.