Numerical PDE solvers outperform neural PDE solvers

TL;DR

DeepFDM is a novel differentiable finite-difference framework that accurately learns PDE coefficients, outperforming neural PDE solvers in stability, efficiency, and interpretability across various scalar PDE benchmarks.

Contribution

Introduces DeepFDM, a stable, interpretable, and efficient finite-difference-based neural framework for learning spatially varying PDE coefficients, outperforming neural PDE solvers.

Findings

DeepFDM achieves 10-20X lower errors than neural PDE solvers.

Requires 10-20X fewer training epochs.

Uses 5-50X fewer parameters.

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.