用可微分有限差分法求解时变偏微分方程，精度和效率远超神经网络方法。
Numerical PDE solvers outperform neural PDE solvers
- 将经典前向欧拉格式嵌入卷积结构，通过满足CFL条件的参数化保证稳定性和一阶收敛。
- 在多类方程上误差比FNO、U-Net等低1到2个数量级，训练次数减少10-20倍，参数量少5-50倍。
- 能准确恢复真实系数场，适合需要可解释性与高精度的物理建模任务。
我们提出DeepFDM，一种用于学习时变偏微分方程中空间变化系数的可微分有限差分框架。通过将经典前向欧拉离散化嵌入卷积架构，DeepFDM利用满足CFL条件的系数参数化实现稳定性与一阶收敛性。模型权重直接对应PDE系数，提供可解释的反问题建模方式。我们在一维、二维和三维空间中的标量PDE基准测试集上评估：包括对流、扩散、对流-扩散、反应-扩散及非齐次Burgers方程。在分布内和分布外测试（以系数先验的Hellinger距离量化）中，DeepFDM的归一化均方误差比Fourier Neural Operators、U-Nets和ResNets低1至2个数量级；训练周期减少10-20倍；参数量减少5-50倍。此外，恢复的系数场与真实参数高度一致。结果表明，DeepFDM是数据驱动求解与识别参数化PDE的稳健、高效且透明的基线方法。
原文摘要 · Abstract (English)
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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