Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds

Abstract

Physics-informed neural networks (PINNs) provide a mesh-free approach to solving high-dimensional PDEs on complex geometries, but their theoretical foundations on manifolds remain limited. Moreover, conventional PINN analyses typically rely on solution smoothness, while PINNs may perform poorly for low-regularity solutions arising from nonlinear hyperbolic equations. In this paper, we develop a weak PINN (wPINN) framework for approximating entropy solutions of geometry-compatible hyperbolic conservation laws on Riemannian manifolds $\mathcal{M}^d$. Building on the well-posedness theory, we establish a localized $L_1$-stability estimate that converts localized entropy residuals into terminal error bounds and leads to a rigorous convergence analysis of the proposed method. We then derive approximation guarantees for time-dependent entropy solutions on manifolds, revealing how approximation errors accumulate over long time horizons. For the quadrature error, we develop a problem-adapted localization complexity analysis and show that, for a fixed adversarial test-network architecture, the solution-network contribution achieves the fast rate $\mathrm{VC}_{\mathcal F}/n$, up to logarithmic factors. The resulting algebraic network-complexity exponent depends only on the intrinsic dimension $d$, rather than the ambient dimension. For fixed localization scales, and up to logarithmic factors and the localization bias, the solution-network statistical exponent matches the corresponding minimax exponent in $d$-dimensional Euclidean Sobolev approximation. Numerical experiments illustrate that the proposed wPINN framework accurately approximates entropy solutions on manifold geometries.