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The curse of dimensionality: what lies beyond the capabilities of physics-informed neural networks

J. Penuela, H. Ouerdane

TL;DR

This paper examines the limits of physics-informed neural networks (PINNs) when solving ill-posed inverse problems by using single- and two-stage RC low-pass filters as a minimal testbed. It shows that PINNs can accurately predict forward dynamics but struggle to uniquely recover multiple unknown parameters in the inverse problem, highlighting identifiability issues when the DE solution has fixed time dimensionality while the parameter space grows. The authors implement a Raissi-style PINN with a physics loss based on RC circuit residuals, generate data with SciPy and validate with PySpice, and perform a HyperOptSearch-based hyperparameter study; results reveal convergence for forward problems but instability and high pole-prediction errors for inverse problems, especially when more than two parameters are inferred. The study delineates practical boundaries for PINN-based parameter discovery and suggests using partial information or priors to enable reliable inverse inference in physical systems.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising framework for solving forward and inverse problems governed by differential equations. However, their reliability when used in ill-posed inverse problems remains poorly understood. In this study, we explore the fundamental limitations of PINNs using a simple illustrative case: RC low-pass filters. Showing that while PINNs can accurately predict system dynamics in forward problems, they fail to recover unique physical parameters when solving inverse problems when more than two parameters are approximated. Our findings provide grounds to understand the boundaries of PINNs applicability for parameter discovery in physical systems.

The curse of dimensionality: what lies beyond the capabilities of physics-informed neural networks

TL;DR

This paper examines the limits of physics-informed neural networks (PINNs) when solving ill-posed inverse problems by using single- and two-stage RC low-pass filters as a minimal testbed. It shows that PINNs can accurately predict forward dynamics but struggle to uniquely recover multiple unknown parameters in the inverse problem, highlighting identifiability issues when the DE solution has fixed time dimensionality while the parameter space grows. The authors implement a Raissi-style PINN with a physics loss based on RC circuit residuals, generate data with SciPy and validate with PySpice, and perform a HyperOptSearch-based hyperparameter study; results reveal convergence for forward problems but instability and high pole-prediction errors for inverse problems, especially when more than two parameters are inferred. The study delineates practical boundaries for PINN-based parameter discovery and suggests using partial information or priors to enable reliable inverse inference in physical systems.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising framework for solving forward and inverse problems governed by differential equations. However, their reliability when used in ill-posed inverse problems remains poorly understood. In this study, we explore the fundamental limitations of PINNs using a simple illustrative case: RC low-pass filters. Showing that while PINNs can accurately predict system dynamics in forward problems, they fail to recover unique physical parameters when solving inverse problems when more than two parameters are approximated. Our findings provide grounds to understand the boundaries of PINNs applicability for parameter discovery in physical systems.

Paper Structure

This paper contains 8 sections, 10 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Method
  4. Dataset
  5. PINN architecture
  6. Hyperparameter optimisation
  7. Results
  8. Discussion and Conclusions

Figures (3)

  • Figure 1: Schematic representation of the simulated low-pass RC filters.
  • Figure 2: Example with 60 samples per cycle from $t=0$ to $t=5$ seconds and the original continuous signal from $t=0$ to $t=10$ seconds for the two-stage RC circuit with the parameters $R_1=15K\Omega$, $C_1=5\mu F$, $R_2=20K\Omega$ and $C_2=15\mu F$.
  • Figure 3: Box plots presenting the distribution $R^2$ against the number of collocation points (C.P.) for the $loss_{phys}$ calculation for the FP after the first iteration of the hyperparameter search.