Solving partial differential equations with sampled neural networks
To address the gradient optimization difficulties and non-causal temporal treatment inherent in physics-informed neural networks (PINNs) for time-dependent partial differential equations (PDEs), this work proposes a gradient-free, causally structured stochastic neural basis function method. Spatially, it constructs neural basis functions with random weights in the hidden layer; temporally, it explicitly integrates time evolution via classical ODE solvers. We introduce a novel dual-mode weight sampling strategy—data-agnostic and data-aware—and establish its $L^2$ convergence in Barron space theoretically. The method combines mesh-free flexibility with spectral convergence accuracy. It enables long-time-domain simulation and inverse problem solving. Numerical experiments across diverse elliptic and time-dependent PDEs demonstrate 1–2 orders-of-magnitude improvements in training speed and accuracy over PINNs, alongside strong generalization capability and numerical stability.