# Physics Informed Constrained Learning of Dynamics from Static Data

arXiv: https://arxiv.org/abs/2504.12675 · HTML: https://mlsanity.com/paper/2504.12675/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 50 references that are in the map.
- First posted on arXiv 17 Apr 2025.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [On the Principles of Parsimony and Self-Consistency for the Emergence of Intelligence](https://mlsanity.com/paper/2207.04630/) (2022)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Belief Propagation Neural Networks](https://mlsanity.com/paper/2007.00295/) (2020)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Systems Biology: Identifiability analysis and parameter identification via systems-biology informed neural networks](https://mlsanity.com/paper/2202.01723/) (2022)
- [Physics-informed Deep Learning for Musculoskeletal Modelling: Predicting Muscle Forces and Joint Kinematics from Surface EMG](https://mlsanity.com/paper/2207.01435/) (2022)
- [A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions](https://mlsanity.com/paper/2102.09924/) (2021)
- [Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics](https://mlsanity.com/paper/2109.00526/) (2021)
- [Hybrid physics-informed metabolic cybergenetics: process rates augmented with machine-learning surrogates informed by flux balance analysis](https://mlsanity.com/paper/2401.00670/) (2024)

## Built on by (most cited first, 1 of 1)

- [A Foundational Multi-Modal Model for Few-Shot Learning](https://mlsanity.com/paper/2508.04746/) (2025)

## Abstract (arXiv)

A physics-informed neural network (PINN) models the dynamics of a system by integrating the governing physical laws into the architecture of a neural network. By enforcing physical laws as constraints, PINN overcomes challenges with data scarsity and potentially high dimensionality. Existing PINN frameworks rely on fully observed time-course data, the acquisition of which could be prohibitive for many systems. In this study, we developed a new PINN learning paradigm, namely Constrained Learning, that enables the approximation of first-order derivatives or motions using non-time course or partially observed data. Computational principles and a general mathematical formulation of Constrained Learning were developed. We further introduced MPOCtrL (Message Passing Optimization-based Constrained Learning) an optimization approach tailored for the Constrained Learning framework that strives to balance the fitting of physical models and observed data. Its code is available at github link: https://github.com/ptdang1001/MPOCtrL Experiments on synthetic and real-world data demonstrated that MPOCtrL can effectively detect the nonlinear dependency between observed data and the underlying physical properties of the system. In particular, on the task of metabolic flux analysis, MPOCtrL outperforms all existing data-driven flux estimators.