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Random Neural Network Expressivity for Non-Linear Partial Differential Equations

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Derives dimension-free N^{-1/2} approximation bounds for random neural networks (RaNNs) on time-dependent Sobolev functions via ridgelet transforms, and applies them to PINN training error for the porous medium and compressible Navier-Stokes equations.

Problem

Approximation/expressivity of randomly weighted one-hidden-layer networks for learning solutions of non-linear time-dependent PDEs, where existing bounds are either fixed-time, require strong decay assumptions, or suffer from curse of dimensionality.

Approach

Uses a ridgelet-transform integral representation and a higher-order Plancherel estimate to show that time-dependent mixed Sobolev functions H^p_t H^q_x can be approximated by an unbiased RaNN with heavy-tailed sampled hidden weights at Monte Carlo rate N^{-1/2} independent of dimension, and propagates this to PINN residual losses.

Key results

• Theory: RaNN unbiased estimator achieves E||u-u_N||^2 <= M_psi/N * ||u||^2, i.e. N^{-1/2} error rate independent of dimension d (dimension enters only via smoothness requirement s_2 > (d+2)/2).
• Numerical experiments for PME in d=1-5 and 1D compressible Navier-Stokes show relative errors close to C/sqrt(N) curves, including with Gaussian weights beyond the theoretical heavy-tailed assumption.
• PINN training residual for PME and Navier-Stokes is bounded by terms of order 1/(N delta) with probability 1-delta.

Surprising findings

• Experiments indicate the N^{-1/2} rate also holds for Gaussian weights, though theory assumes heavy-tailed weight distributions.

Limitations

• Theory requires smooth solutions (high Sobolev regularity) and a heavy-tailed weight distribution, while experiments use Gaussian weights.
• Constant in the bound not explicit and may grow with dimension; only Lipschitz bounded domains and 1D periodic Navier-Stokes considered.
• Numerical validation limited to one PDE with a known analytic solution per setting (PME with m=2, 1D NS shock).

Experimental setup

Datasets: Barenblatt-Kompaneets-Zeldovich PME solution compressible Navier-Stokes travelling shock-wave solution

Benchmarks: PME (m=2) in d=1-5 with Barenblatt solution compressible Navier-Stokes 1D travelling shock wave

Metrics: relative L2 error PINN residual loss bound

Baselines: Barenblatt-Kompaneets-Zeldovich solution (reference) travelling shock-wave solution (reference)

Builds on: PINNs ridgelet transform random feature models Barron-type approximation

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