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

arXiv: https://arxiv.org/abs/2505.19036 · HTML: https://mlsanity.com/paper/2505.19036/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2025.

## Built on by (most cited first, 3 of 3)

- [Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation](https://mlsanity.com/paper/2509.09362/) (2025)
- [Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems](https://mlsanity.com/paper/2605.04126/) (2026)
- [Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws](https://mlsanity.com/paper/2608.10389/) (2026)

## Abstract (arXiv)

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 Md. Building on the well-posedness theory, we establish a localized L1-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 VCmathcal 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.