# Coercing Machine Learning to Output Physically Accurate Results

arXiv: https://arxiv.org/abs/1910.09671 · HTML: https://mlsanity.com/paper/1910.09671/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2019.

## Built on by (most cited first, 10 of 10)

- [Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data](https://mlsanity.com/paper/2206.06577/) (2022)
- [Virtual Elastic Objects](https://mlsanity.com/paper/2201.04623/) (2022)
- [ElastoGen: 4D Generative Elastodynamics](https://mlsanity.com/paper/2405.15056/) (2024)
- [Constrained Adversarial Learning for Automated Software Testing: a literature review](https://mlsanity.com/paper/2303.07546/) (2023)
- [SDYN-GANs: Adversarial Learning Methods for Multistep Generative Models for General Order Stochastic Dynamics](https://mlsanity.com/paper/2302.03663/) (2023)
- [Analytically Integratable Zero-restlength Springs for Capturing Dynamic Modes unrepresented by Quasistatic Neural Networks](https://mlsanity.com/paper/2201.10122/) (2022)
- [PhysXNet: A Customizable Approach for LearningCloth Dynamics on Dressed People](https://mlsanity.com/paper/2111.07195/) (2021)
- [Neural Partial Differential Equations with Functional Convolution](https://mlsanity.com/paper/2303.07194/) (2023)
- [A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems](https://mlsanity.com/paper/2503.15441/) (2025)
- [Addressing Discontinuous Root-Finding for Subsequent Differentiability in Machine Learning, Inverse Problems, and Control](https://mlsanity.com/paper/2306.12413/) (2023)

## Abstract (arXiv)

Many machine/deep learning artificial neural networks are trained to simply be interpolation functions that map input variables to output values interpolated from the training data in a linear/nonlinear fashion. Even when the input/output pairs of the training data are physically accurate (e.g. the results of an experiment or numerical simulation), interpolated quantities can deviate quite far from being physically accurate. Although one could project the output of a network into a physically feasible region, such a postprocess is not captured by the energy function minimized when training the network; thus, the final projected result could incorrectly deviate quite far from the training data. We propose folding any such projection or postprocess directly into the network so that the final result is correctly compared to the training data by the energy function. Although we propose a general approach, we illustrate its efficacy on a specific convolutional neural network that takes in human pose parameters (joint rotations) and outputs a prediction of vertex positions representing a triangulated cloth mesh. While the original network outputs vertex positions with erroneously high stretching and compression energies, the new network trained with our physics prior remedies these issues producing highly improved results.