Random Neural Network Expressivity for Non-Linear Partial Differential Equations
For researchers in scientific computing and machine learning, this provides theoretical guarantees for using RaNNs to solve non-linear PDEs, though the results are incremental as they extend existing RaNN theory to a new problem domain.
The authors derive error bounds for random neural network (RaNN) approximations to time-dependent Sobolev functions, achieving a dimension-free approximation rate of 1/2 for sufficiently regular functions, and demonstrate efficient approximation of solutions to non-linear PDEs such as Porous Medium and Compressible Navier-Stokes equations.
Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.