Integrating Physics-Informed Deep Learning and Numerical Methods for Robust Dynamics Discovery and Parameter Estimation

TL;DR

This work combines deep learning techniques and classic numerical methods for differential equations to solve two challenging problems in dynamical systems theory: dynamics discovery and parameter estimation.

Abstract

Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to solve two challenging problems in dynamical systems theory: dynamics discovery and parameter estimation. Results demonstrate the effectiveness of the proposed approaches on a suite of test problems exhibiting oscillatory and chaotic dynamics. When comparing the performance of various numerical schemes, such as the Runge-Kutta and linear multistep families of methods, we observe promising results in predicting the system dynamics and estimating physical parameters, given appropriate choices of spatial and temporal discretization schemes and numerical method orders.

Figures (11)

• Figure 1: Schematic for the proposed neural network with numerical ODE methods to model dynamical systems with completely unknown dynamics. We obtain an approximation for the RHS dynamics $\hat{\mathbf{f}}_{\textit{DNN}}$ using the observed states as input, then use $\hat{\mathbf{f}}_{\textit{DNN}}$ to compute a numerical approximation of the states $\Vec{X}_{\textit{NM}}$. The observed and approximated states are used in the loss function to update $\hat{\mathbf{f}}_{\textit{DNN}}$.
• Figure 2: Dynamics discovery model predictions of the FitzHugh-Nagumo model obtained using observed data at various noise levels ($\delta = 0, 0.1, 0.2$) with RK45 and BDF2.
• Figure 3: Parameter estimation predictions of the FitzHugh-Nagumo model states obtained using observed data at various noise levels ($\delta = 0, 0.2$) with RK45.
• Figure 4: Parameter estimation model predictions of FitzHugh-Nagumo model with 20% noisy data for RK (RK45) and LMM (AB2) schemes without pre-training.
• Figure 5: Dynamics discovery model predictions of the Lorenz-63 system obtained using noisy observed data at various noise levels ($\delta = 0, 0.1, 0.2$) with BDF2.