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Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI

Yoh-ichi Mototake, Makoto Sasaki

TL;DR

This work addresses the non-uniqueness and uncertain interpretability that can arise when estimating PDE coefficient functions within physics-informed ML. It introduces a rank-based, three-step framework to quantify and reduce structural uncertainty, integrating physical constraints to achieve unique identification where possible. Demonstrations on the wave kinetic equation show that symmetry constraints can convert an underdetermined problem into a uniquely identifiable one, though hyperparameter choices based on predictive loss alone can be misleading. The approach provides a rigorous pathway to physically meaningful inference in data-rich scientific settings and suggests avenues for extending uncertainty analysis to broader PDE classes.

Abstract

Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data-driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data-driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.

Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI

TL;DR

This work addresses the non-uniqueness and uncertain interpretability that can arise when estimating PDE coefficient functions within physics-informed ML. It introduces a rank-based, three-step framework to quantify and reduce structural uncertainty, integrating physical constraints to achieve unique identification where possible. Demonstrations on the wave kinetic equation show that symmetry constraints can convert an underdetermined problem into a uniquely identifiable one, though hyperparameter choices based on predictive loss alone can be misleading. The approach provides a rigorous pathway to physically meaningful inference in data-rich scientific settings and suggests avenues for extending uncertainty analysis to broader PDE classes.

Abstract

Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data-driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data-driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.

Paper Structure

This paper contains 16 sections, 1 theorem, 27 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. PIML and Inverse Problem
  4. Uncertainty in Inverse Problems
  5. Theoretical Preliminaries for Uncertainty Evaluation
  6. Examples
  7. Proposed Framework: PIML with Uncertainty Evaluation
  8. Demonstration
  9. Wave Kinetic Equation
  10. Uncertainty Evaluation ( Step 1)
  11. Introduce Physical Constraints ( Step 2)
  12. Estimation Results of Hamiltonian for Each Hyperparameter $\lambda$ ( Step 3)
  13. Summary and Discussion
  14. Appendix: DNN Model and its Training Parameters
  15. Bridging the Assumption of Infinitesimal Grids in Uncertainty Evaluation and Learning with Finite Data using DNN Model
  16. ...and 1 more sections

Key Result

Theorem 1

Let $u : \mathbb{R}^d \to \mathbb{R}$ and $a : \mathbb{R}^d \to \mathbb{R}$ be a sufficiently smooth function, and consideration a $m$-th PDEs of the form where $A_{\geq m} := \{\alpha \mid m \leq |\alpha|\}$, $m \geq k$, and $\partial^k u(x)$ represents the arbitrary set of $k$-th-order partial differential coefficients. That is, PDEs is linear in $\partial^\alpha a(x)$. Assume that PDEs have a

Figures (6)

  • Figure 1: Risks of scientific research using machine learning. In the presence of an uncertainty, a machine learning model may sometimes provide an interpretation that is physically unfavorable. (The figure was generated using DALL-E3, OpenAI)
  • Figure 2: Conceptual diagram of Hamiltonian function estimation based on the wave kinetic equation.
  • Figure 3: (a) Hamiltonian function $H(x,k_x)$ set up in the simulation. (b) Turbulence intensity data $I(x,k_x)$ obtained from the simulation.
  • Figure 4: (a) Estimation results for the Hamiltonian function $H_{\theta_{\rm dnn}}(x,k_x)$ with symmetry constraints and (b) without constraints. The histogram represents the DNN function estimation results, and the red contour line represents the Hamiltonian function set when generating the dataset.
  • Figure 5: Conceptual diagram of different integration paths in $(x,k_x)$ space to obtain the coefficient function. The red path to integrate $k_x$ first and the blue path to integrate $x$ are shown.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Definition 1: $k$-Leaf Set of Partial Derivatives
  • Theorem 1: Uniqueness of Coefficient Function up to Polynomial under Root Derivative Information