Multiscale Regularization of Physics-Informed Neural Networks Based on a Maximal Local Oscillation Operator for Problems with Sharp Gradients

This paper proposes a multiscale regularizer for physics-informed neural networks (PINNs) designed for differential equation problems with sharp spatial gradients. The regularizer is based on a maximal local oscillation operator that measures the deviation of a function from its local mean across several scales. Unlike standard penalties on high-order derivatives, the proposed approach does not require their explicit computation and naturally accounts for the multiscale structure of the solution. For practical use, we introduce a differentiable discrete implementation of the operator on a finite set of radii using one-dimensional convolutions, compatible with automatic differentiation in PyTorch. The method is integrated into the PINN loss function as an additional regularizer that suppresses local nonphysical oscillations. Using the viscous Burgers equation as an example, we compare the method against a standard PINN and several classical smoothness penalties at several viscosity values; the exact solution obtained via the Cole–Hopf transformation serves as the reference. Numerical results show that, in the front-formation regime, the proposed approach achieves the best mean approximation accuracy among the methods considered while preserving the sharpness of the front and avoiding the over-smoothing characteristic of penalties on high-order derivatives; in the extremely smooth and extremely stiff regimes the method is on par with the alternatives.