Physics Informed Random Feature Neural Networks for Solving PDEs

We present an improved neural field architecture for solving partial differential equations (PDEs). Current physics-informed neural networks (PINNs) provide a flexible framework for solving PDEs, but they struggle to achieve highly accurate solutions and require computation that scales poorly with parameter count. Our model, which we call beignet (Bandlimited Embedding with Interpolated Grid Network), replaces the random Fourier feature embedding used by existing PINN models with a trainable multi-resolution Fourier feature pyramid. To query beignet at a continuous coordinate, we use Fourier interpolation at each level of the pyramid to return features at the input coordinate, and then decode this vector with a fully-connected neural network trunk. Our model provides multiple benefits: 1) Spatial derivatives can be computed efficiently by using the chain rule to compose derivatives of the neural network computed with automatic differentiation with derivatives of the feature grid computed spectrally by the Fast Fourier transform (FFT). 2) beignet can achieve higher accuracy in a compute-efficient manner by scaling the parameter count of this Fourier feature pyramid, instead of the less-efficient strategy of scaling the neural network architecture. 3) beignet can directly control the representation bandlimit, resulting in more stable optimization for difficult PDEs. We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state-of-the-art PINN methods. We further evaluate beignet on the self-similar inviscid Burgers blowup problem and show that it can minimize residuals to near machine precision using Adam, an accuracy regime previously attained only by using computationally expensive higher-order optimizers.

Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to $23\times$ fewer collocation points and a vanilla architecture.

Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative $L^2$ errors up to two orders of magnitude lower than state-of-the-art baselines.