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Structural Constraints for Physics-augmented Learning

Simon Kuang, Xinfan Lin

TL;DR

Two criteria that can be used to assert integrity that a hybrid (physics plus black-box) model should be unable to replicate the physical model are proposed and any best-fit hybrid model has the same physical parameter as a best-fit standalone physics model.

Abstract

When the physics is wrong, physics-informed machine learning becomes physics-misinformed machine learning. A powerful black-box model should not be able to conceal misconceived physics. We propose two criteria that can be used to assert integrity that a hybrid (physics plus black-box) model: 0) the black-box model should be unable to replicate the physical model, and 1) any best-fit hybrid model has the same physical parameter as a best-fit standalone physics model. We demonstrate them for a sample nonlinear mechanical system approximated by its small-signal linearization.

Structural Constraints for Physics-augmented Learning

TL;DR

Two criteria that can be used to assert integrity that a hybrid (physics plus black-box) model should be unable to replicate the physical model are proposed and any best-fit hybrid model has the same physical parameter as a best-fit standalone physics model.

Abstract

When the physics is wrong, physics-informed machine learning becomes physics-misinformed machine learning. A powerful black-box model should not be able to conceal misconceived physics. We propose two criteria that can be used to assert integrity that a hybrid (physics plus black-box) model: 0) the black-box model should be unable to replicate the physical model, and 1) any best-fit hybrid model has the same physical parameter as a best-fit standalone physics model. We demonstrate them for a sample nonlinear mechanical system approximated by its small-signal linearization.
Paper Structure (15 sections, 3 theorems, 20 equations, 6 figures, 1 table)

This paper contains 15 sections, 3 theorems, 20 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model structure
  3. Motivation for PAML
  4. Principles for PAML
  5. Contributions
  6. Notation
  7. Method of incompatibility certificates
  8. Motivation: harmonics of a nonlinear system
  9. Definitions
  10. Consequences
  11. Example
  12. Residual model specification
  13. Numerical results
  14. Conclusion
  15. Methods

Key Result

Theorem 1

Suppose furthermore that $\mathcal{H} \perp_u \mathcal{R}$. Then for all$\hat{\theta} \in {\Theta}$,

Figures (6)

  • Figure 1: Real trajectory and physical model identification at $\alpha = 4$. Mismatch norm is 12.01%.
  • Figure 2: Real trajectory and residual model identification at $\alpha = 4$. Physical model mismatch norm is 12.08%; hybrid model mismatch norm is 5.24%.
  • Figure 3: Real trajectory and hybrid model on a pseudorandom validation input at $\alpha = 4$. Physical model mismatch norm is 12.45%; hybrid model mismatch norm is 6.25%.
  • Figure 4: Real trajectory and physical model identification at $\alpha = 10$. Mismatch norm is 90.71%.
  • Figure 5: Real trajectory and residual model identification at $\alpha = 10$. Physical model mismatch norm is 99.84%; hybrid model mismatch norm is 76.42%.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 1: Zeroth-order incompatibility witness
  • Definition 2: First-order incompatibility witness
  • proof
  • Theorem 1: Zeroth-order incompatibility
  • proof
  • proof
  • Theorem 2: First-order incompatibility
  • proof
  • Proposition 1
  • proof