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A novel framework for generalization of deep hidden physics models

Vijay Kag, Birupaksha Pal

TL;DR

This work tackles the challenge of modeling systems with incomplete physics while requiring generalization to changing inputs, parameters, and domains. It proposes a deep hidden physics model (DHPM) built from a pair of networks (N_sol for the state and N_hid for hidden dynamics) connected through a residual term g(x,t) that enforces the partially known PDE, with total loss L_total = L_data + L_equation and L_equation = (1/M) Σ |g(x_i,t_i)|^2. The authors demonstrate three generalization axes: (i) input generalization by discretizing and feeding input functions, (ii) parameter generalization by adding system parameters as inputs, and (iii) domain generalization by including domain geometry (length L) as an input feature. Across reaction-diffusion experiments, the framework shows robust prediction for unseen inputs, unseen parameter values, and unseen domain configurations, while optionally revealing the learned hidden physics. The approach offers a data-efficient, interpretable pathway for system identification and potential discovery of governing dynamics, with practical implications for industrial modeling and digital-twin development; it can be extended to Gaussian processes or alternative surrogate models.

Abstract

Modelling of systems where the full system information is unknown is an oft encountered problem for various engineering and industrial applications, as it's either impossible to consider all the complex physics involved or simpler models are considered to keep within the limits of the available resources. Recent advances in greybox modelling like the deep hidden physics models address this space by combining data and physics. However, for most real-life applications, model generalizability is a key issue, as retraining a model for every small change in system inputs and parameters or modification in domain configuration can render the model economically unviable. In this work we present a novel enhancement to the idea of hidden physics models which can generalize for changes in system inputs, parameters and domains. We also show that this approach holds promise in system discovery as well and helps learn the hidden physics for the changed system inputs, parameters and domain configuration.

A novel framework for generalization of deep hidden physics models

TL;DR

This work tackles the challenge of modeling systems with incomplete physics while requiring generalization to changing inputs, parameters, and domains. It proposes a deep hidden physics model (DHPM) built from a pair of networks (N_sol for the state and N_hid for hidden dynamics) connected through a residual term g(x,t) that enforces the partially known PDE, with total loss L_total = L_data + L_equation and L_equation = (1/M) Σ |g(x_i,t_i)|^2. The authors demonstrate three generalization axes: (i) input generalization by discretizing and feeding input functions, (ii) parameter generalization by adding system parameters as inputs, and (iii) domain generalization by including domain geometry (length L) as an input feature. Across reaction-diffusion experiments, the framework shows robust prediction for unseen inputs, unseen parameter values, and unseen domain configurations, while optionally revealing the learned hidden physics. The approach offers a data-efficient, interpretable pathway for system identification and potential discovery of governing dynamics, with practical implications for industrial modeling and digital-twin development; it can be extended to Gaussian processes or alternative surrogate models.

Abstract

Modelling of systems where the full system information is unknown is an oft encountered problem for various engineering and industrial applications, as it's either impossible to consider all the complex physics involved or simpler models are considered to keep within the limits of the available resources. Recent advances in greybox modelling like the deep hidden physics models address this space by combining data and physics. However, for most real-life applications, model generalizability is a key issue, as retraining a model for every small change in system inputs and parameters or modification in domain configuration can render the model economically unviable. In this work we present a novel enhancement to the idea of hidden physics models which can generalize for changes in system inputs, parameters and domains. We also show that this approach holds promise in system discovery as well and helps learn the hidden physics for the changed system inputs, parameters and domain configuration.
Paper Structure (22 sections, 10 equations, 38 figures)

This paper contains 22 sections, 10 equations, 38 figures.

Table of Contents

  1. Introduction
  2. Deep Hidden Physics Models
  3. Generalization for inputs
  4. Prediction for similar kind of unseen input function
  5. Prediction for out of distribution input functions
  6. Quadratic function
  7. Cubic function
  8. Trigonometric function
  9. Generalization of parameters
  10. Unseen parameter and similar kind of unseen input function
  11. Prediction for unseen parameter and out of distribution input functions
  12. Unseen parameter and unseen quadratic input
  13. Unseen parameter and unseen cubic input
  14. Unseen parameter and unseen trigonometric input
  15. Error analysis for different kind of input functions
  16. ...and 7 more sections

Figures (38)

  • Figure 1: Networks for the solution and the hidden physics
  • Figure 2: Deep hidden Physics model framework
  • Figure 3: Network architecture for generalization of input
  • Figure 4: A few random input functions
  • Figure 5: Error corresponding to number of data points, The train error is calculated by taking mean of $L_2 \ \text{Error}$ of solutions predicted for $n_{fun}$ number of input function that were used for training and similarly the test error is calculated by taking mean of $L_2 \ \text{Error}$ of solutions predicted for 1000 random input functions of the same form given in \ref{['periodic_input_function']}.
  • ...and 33 more figures