{"ID":2889772,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.21269","arxiv_id":"2507.21269","title":"Numerical PDE solvers outperform neural PDE solvers","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.","short_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-co...","url_abs":"https://arxiv.org/abs/2507.21269","url_pdf":"https://arxiv.org/pdf/2507.21269v1","authors":"[\"Patrick Chatain\",\"Michael Rizvi-Martel\",\"Guillaume Rabusseau\",\"Adam Oberman\"]","published":"2025-07-28T18:50:37Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}