# Physics-consistent machine learning: output projection onto physical manifolds

arXiv: https://arxiv.org/abs/2502.15755 · HTML: https://mlsanity.com/paper/2502.15755/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 4 of 4)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Deep neural networks for the evaluation and design of photonic devices](https://mlsanity.com/paper/2007.00084/) (2020)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Efficient Learning of Accurate Surrogates for Simulations of Complex Systems](https://mlsanity.com/paper/2207.12855/) (2022)

## Built on by (most cited first, 6 of 6)

- [A graph neural network surrogate model for mesh-based crashworthiness prediction of vehicle panel components](https://mlsanity.com/paper/2503.17386/) (2025)
- [Divergence-Free Diffusion Models for Incompressible Fluid Flows](https://mlsanity.com/paper/2601.19368/) (2026)
- [Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution](https://mlsanity.com/paper/2603.09693/) (2026)
- [Conditional Denoising Model as a Physical Surrogate Model](https://mlsanity.com/paper/2601.21021/) (2026)
- [Multi-Label Phase Diagram Prediction in Complex Alloys via Physics-Informed Graph Attention Networks](https://mlsanity.com/paper/2604.16468/) (2026)
- [Quantum Reservoir Computing with Physics-Informed Correction for Reduced-Order PDE Forecasting](https://mlsanity.com/paper/2608.23119/) (2026)

## Abstract (arXiv)

Data-driven machine learning models often require extensive datasets, which can be costly or inaccessible, and their predictions may fail to comply with established physical laws. Current approaches for incorporating physical priors mitigate these issues by penalizing deviations from known physical laws, as in physics-informed neural networks, or by designing architectures that automatically satisfy specific invariants. However, penalization approaches do not guarantee compliance with physical constraints for unseen inputs, and invariant-based methods lack flexibility and generality. We propose a novel physics-consistent machine learning method that directly enforces compliance with physical principles by projecting model outputs onto the manifold defined by these laws. This procedure ensures that predictions inherently adhere to the chosen physical constraints, improving reliability and interpretability. Our method is demonstrated on two systems: a spring-mass system and a low-temperature reactive plasma. Compared to purely data-driven models, our approach significantly reduces errors in physical law compliance, enhances predictive accuracy of physical quantities, and outperforms alternatives when working with simpler models or limited datasets. The proposed projection-based technique is versatile and can function independently or in conjunction with existing physics-informed neural networks, offering a powerful, general, and scalable solution for developing fast and reliable surrogate models of complex physical systems, particularly in resource-constrained scenarios.