Numerical PDE solvers outperform neural PDE solvers

• URL: http://arxiv.org/abs/2507.21269v1
• Date: Mon, 28 Jul 2025 18:50:37 GMT
• Title: Numerical PDE solvers outperform neural PDE solvers
• Authors: Patrick Chatain, Michael Rizvi-Martel, Guillaume Rabusseau, Adam Oberman,
• Abstract summary: DeepFDM is a finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations.<n>It enforces stability and first-order convergence via CFL-compliant coefficient parameterizations.<n>It attains normalized mean-squared errors one to two orders of magnitude smaller than Fourier Neural Operators, U-Nets and ResNets.
• Score: 5.303553599778495
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional architecture, DeepFDM enforces stability and first-order convergence via CFL-compliant coefficient parameterizations. Model weights correspond directly to PDE coefficients, yielding an interpretable inverse-problem formulation. We evaluate DeepFDM on a benchmark suite of scalar PDEs: advection, diffusion, advection-diffusion, reaction-diffusion and inhomogeneous Burgers' equations-in one, two and three spatial dimensions. In both in-distribution and out-of-distribution tests (quantified by the Hellinger distance between coefficient priors), DeepFDM attains normalized mean-squared errors one to two orders of magnitude smaller than Fourier Neural Operators, U-Nets and ResNets; requires 10-20X fewer training epochs; and uses 5-50X fewer parameters. Moreover, recovered coefficient fields accurately match ground-truth parameters. These results establish DeepFDM as a robust, efficient, and transparent baseline for data-driven solution and identification of parametric PDEs.

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