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Physics Informed Constrained Learning of Dynamics from Static Data

Pengtao Dang, Tingbo Guo, Melissa Fishel, Guang Lin, Wenzhuo Wu, Sha Cao, Chi Zhang

TL;DR

This work extends physics-informed learning to scenarios where time-series or complete variable observations are unavailable. By formulating Constrained Learning on a directed factor-graph representation and introducing MPOCtrL, the authors jointly enforce physical laws via a coherency loss and promote parsimonious, interpretable flux representations through a gating-enabled, BP-inspired optimization. Empirical results on synthetic and real metabolic networks demonstrate accurate flux estimation from non-time-course data and robustness to noise, outperforming state-of-the-art data-driven flux estimators. The approach broadens the applicability of physics-informed methods to static or sparse observational settings with potential impact in systems biology and network dynamics.

Abstract

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.

Physics Informed Constrained Learning of Dynamics from Static Data

TL;DR

This work extends physics-informed learning to scenarios where time-series or complete variable observations are unavailable. By formulating Constrained Learning on a directed factor-graph representation and introducing MPOCtrL, the authors jointly enforce physical laws via a coherency loss and promote parsimonious, interpretable flux representations through a gating-enabled, BP-inspired optimization. Empirical results on synthetic and real metabolic networks demonstrate accurate flux estimation from non-time-course data and robustness to noise, outperforming state-of-the-art data-driven flux estimators. The approach broadens the applicability of physics-informed methods to static or sparse observational settings with potential impact in systems biology and network dynamics.

Abstract

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.

Paper Structure

This paper contains 26 sections, 10 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Physics Informed Neural Networks (PINNs)
  4. Data Driven Flux Analysis
  5. Mathematical consideration and method formulations
  6. Notations and preliminaries
  7. Constrained Learning to solve PINN with non-time course and partially observed data
  8. A new optimization strategy to solve the general Constrained Learning problem
  9. Estimating flux over a network by Constrained Learning
  10. Experiments
  11. Experimental setup
  12. Experiments on Synthetic Observation Data
  13. Experiments on Real-world Observation Data
  14. Conclusion
  15. Appendix / supplemental material
  16. ...and 11 more sections

Figures (11)

  • Figure 1: (a) Constrained Learning-based formulation of the flux estimation problem, (b) geometric illustration of the MPOCtrL optimization algorithm, (c) framework of the MPOCtrL algorithm.
  • Figure 2: Experiments of MPOCtrL and MPO on Synthetic Data and Real-world Data
  • Figure A.1: MPO Operations Example
  • Figure A.2: Experiments about Effectiveness, Robustness, Running Time of MPO on four Directed Factor Graphs
  • Figure A.3: Comparisons between MPO and BRW on four directed factor graphs for different error level gammas and running time
  • ...and 6 more figures

Theorems & Definitions (2)

  • Definition 1
  • Definition 2