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Conformalized Physics-Informed Neural Networks

Lena Podina, Mahdi Torabi Rad, Mohammad Kohandel

TL;DR

The paper introduces Conformalized PINNs (C-PINNs) to quantify uncertainty in physics-informed neural networks without relying on distributional assumptions, by leveraging Split Conformal prediction to produce finite-sample uncertainty intervals with coverage guarantees. It demonstrates forward uncertainty quantification for the logistic-growth ODE and a Buckley-Leverett PDE (including a discontinuous solution) and inverse uncertainty quantification for a parameter $\beta$, across noiseless and noisy data. The method uses calibration and validation splits to compute nonconformity scores and interval quantiles, ensuring coverage at level $1-\alpha$. The work discusses averaging over splits to verify exact coverage and outlines potential extensions to UPINNs and adaptive conformal intervals for enhanced efficiency and applicability.

Abstract

Physics-informed neural networks (PINNs) are an influential method of solving differential equations and estimating their parameters given data. However, since they make use of neural networks, they provide only a point estimate of differential equation parameters, as well as the solution at any given point, without any measure of uncertainty. Ensemble and Bayesian methods have been previously applied to quantify the uncertainty of PINNs, but these methods may require making strong assumptions on the data-generating process, and can be computationally expensive. Here, we introduce Conformalized PINNs (C-PINNs) that, without making any additional assumptions, utilize the framework of conformal prediction to quantify the uncertainty of PINNs by providing intervals that have finite-sample, distribution-free statistical validity.

Conformalized Physics-Informed Neural Networks

TL;DR

The paper introduces Conformalized PINNs (C-PINNs) to quantify uncertainty in physics-informed neural networks without relying on distributional assumptions, by leveraging Split Conformal prediction to produce finite-sample uncertainty intervals with coverage guarantees. It demonstrates forward uncertainty quantification for the logistic-growth ODE and a Buckley-Leverett PDE (including a discontinuous solution) and inverse uncertainty quantification for a parameter , across noiseless and noisy data. The method uses calibration and validation splits to compute nonconformity scores and interval quantiles, ensuring coverage at level . The work discusses averaging over splits to verify exact coverage and outlines potential extensions to UPINNs and adaptive conformal intervals for enhanced efficiency and applicability.

Abstract

Physics-informed neural networks (PINNs) are an influential method of solving differential equations and estimating their parameters given data. However, since they make use of neural networks, they provide only a point estimate of differential equation parameters, as well as the solution at any given point, without any measure of uncertainty. Ensemble and Bayesian methods have been previously applied to quantify the uncertainty of PINNs, but these methods may require making strong assumptions on the data-generating process, and can be computationally expensive. Here, we introduce Conformalized PINNs (C-PINNs) that, without making any additional assumptions, utilize the framework of conformal prediction to quantify the uncertainty of PINNs by providing intervals that have finite-sample, distribution-free statistical validity.
Paper Structure (12 sections, 2 equations, 8 figures)

This paper contains 12 sections, 2 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Methodology and Results
  3. Conformalizing the PINN surrogate solution (forward problem)
  4. Conformalizing the logistic growth ODE
  5. Conformalizing a PDE with a discontinous solution (Buckley-Leverett Equation)
  6. Conformalizing an ODE parameter estimate (inverse problem)
  7. Conclusion and Discussion
  8. Appendix
  9. Conformalized forward problem: Noiseless data
  10. Model architecture and training
  11. Conformalized inverse problem: noisy data
  12. Conformalized inverse problem: repeated validation/calibration splits

Figures (8)

  • Figure 1: (a) The performance of a single PINN on noisy data (noise level 0.08). (b) Using the PINN and noisy data from Fig 1a, coverage estimated through repeated construction of an interval and validation, for different calibration and validation sets.
  • Figure 2: (a) The performance of a single PINN in approximating the Buckley-Leverett equation (eq. \ref{['eq:bl']}). (b) Using the PINN and data from Fig 2a, coverage estimated through repeated construction of an interval and validation, for different calibration and validation sets.
  • Figure 3: (a) Plot of the inferred $\hat{\beta}$ vs. true $\beta$ for 1000 datasets generated from sampled values of $\beta$ (noiseless data) (b) Estimated coverage using the holdout set of $\{\beta, \hat{\beta}\}$ from Fig \ref{['fig:cal_alphas']}.
  • Figure 4: (a) Plot of the inferred vs. true $\beta$ for 1000 datasets generated from sampled values of $\beta$ (data noise level is 0.03) (b) Estimated coverage using the holdout set of $\{\beta, \hat{\beta}\}$ from Fig 4a.
  • Figure 5: (a) The performance of a single PINN on non-noisy data. (b) Using the PINN and noiseless data from Fig \ref{['fig:noiseless_pinn']}, coverage estimated through repeated construction of an interval, for different calibration and validation sets. The running average coverage is computed over 10000 trials.
  • ...and 3 more figures