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Towards Model Discovery Using Domain Decomposition and PINNs

Tirtho S. Saha, Alexander Heinlein, Cordula Reisch

TL;DR

A better performance is found for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.

Abstract

We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi-stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, we find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.

Towards Model Discovery Using Domain Decomposition and PINNs

TL;DR

A better performance is found for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.

Abstract

We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi-stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, we find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.
Paper Structure (13 sections, 18 equations, 8 figures, 3 tables)

This paper contains 13 sections, 18 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Computational Methods
  3. Physics Informed Neural Networks(PINNs)
  4. Domain Decomposition-Based PINNs(FBPINNs)
  5. Mathematical Models
  6. Saturated growth model
  7. Competition Model
  8. Model discovery and parameter estimation
  9. Results
  10. Model accuracy and parameter learning
  11. Loss landscapes with and without domain decomposition
  12. Noise Effect
  13. Training parameters

Figures (8)

  • Figure 1: Window function \ref{['eq:SubdomainWindowFunction']}. The light blue and light orange regions represent the respective subdomain intervals, while the combined light brown region highlights the overlap between the two subdomains.
  • Figure 2: Solutions of (a) the saturated growth model \ref{['eq:SaturatedGrowthModel']} and (b) the competition model \ref{['eq:CompetitionModel']} for parameters with coexistence ($u_c, v_c)$ or single-survival $(u_s, v_s)$. The dynamic time frame is shaded in blue, the quasi-stationary in gray.
  • Figure 3: Learned parameters in the competition model. The values $[a,b]$ give the time domain of data used.
  • Figure 4: Energy plots for the competition model \ref{['eq:CompetitionModel']} in the coexistence setting with data in $[0, 24]$.
  • Figure 5: MSE of the vanilla PINN solution and the FBPINN solution based on data from three time intervals.
  • ...and 3 more figures