# Learning Under Laws: A Constraint-Projected Neural PDE Solver that Eliminates Hallucinations

arXiv: https://arxiv.org/abs/2511.03578 · HTML: https://mlsanity.com/paper/2511.03578/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 13 references that are in the map.
- First posted on arXiv 5 Nov 2025.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)

## Built on by (most cited first, 2 of 2)

- [FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates](https://mlsanity.com/paper/2602.01941/) (2026)
- [A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers](https://mlsanity.com/paper/2609.07437/) (2026)

## Abstract (arXiv)

Neural networks can approximate solutions to partial differential equations, but they often break the very laws they are meant to model-creating mass from nowhere, drifting shocks, or violating conservation and entropy. We address this by training within the laws of physics rather than beside them. Our framework, called Constraint-Projected Learning (CPL), keeps every update physically admissible by projecting network outputs onto the intersection of constraint sets defined by conservation, Rankine-Hugoniot balance, entropy, and positivity. The projection is differentiable and adds only about 10% computational overhead, making it fully compatible with back-propagation. We further stabilize training with total-variation damping (TVD) to suppress small oscillations and a rollout curriculum that enforces consistency over long prediction horizons. Together, these mechanisms eliminate both hard and soft violations: conservation holds at machine precision, total-variation growth vanishes, and entropy and error remain bounded. On Burgers and Euler systems, CPL produces stable, physically lawful solutions without loss of accuracy. Instead of hoping neural solvers will respect physics, CPL makes that behavior an intrinsic property of the learning process.