Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations

This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying results against noisy data, partly owing to suboptimal estimated derivatives and found PDE coefficients. We address the issues by introducing a noise-aware physics-informed machine learning (nPIML) framework to discover the governing PDE from data following arbitrary distributions. We propose training a couple of neural networks, namely solver and preselector, in a multi-task learning paradigm, which yields important scores of basis candidates that constitute the hidden physical constraint. After they are jointly trained, the solver network estimates potential candidates, e.g., partial derivatives, for the sparse regression to initially unveil the most likely parsimonious PDE, decided according to information criterion. Denoising physics-informed neural networks (dPINNs), based on Discrete Fourier Transform (DFT), is proposed to deliver the optimal PDE coefficients respecting the noise-reduced variables. Extensive experiments on five canonical PDEs affirm that the proposed framework presents a robust and interpretable approach for PDE discovery, leading to a new automatic PDE selection algorithm established on minimization of the information criterion decay rate.

Section: Derivative Preparationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Noise-aware physics-informed machine learning for robust PDE discovery

Thanasutives

Morita

Numao

et al. 2023

“…The broad utility of PINNs is revealed by their numerous applications to problems in aerodynamics [16], surface physics [17], power systems [18], cardiology [19], and soft biological tissues [20]. PINNs have also been integrated into the multi-task learning [21] and meta-learning [22] frameworks. When implementing PINN algorithms to find functions in an unbounded system, the unbounded variables cannot be simply normalized, precluding reconstruction of solutions outside the range of data.…”

Section: Introductionmentioning

confidence: 99%

Spectrally adapted physics-informed neural networks for solving unbounded domain problems

Xia

Böttcher

Chou

2023

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient numerical methods that can resolve the dependence of the PDE on the unbounded variable over at least several orders of magnitude. We propose a solution to such problems by combining two classes of numerical methods: (i) adaptive spectral methods and (ii) physics-informed neural networks (PINNs). The numerical approach that we develop takes advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs and extrapolate numerical solutions at any point in space and time. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs in solving PDEs and estimating model parameters from noisy observations in unbounded domains.

“…However, due to inaccurate coefficient optimization, this method may recognise errors. Thanasutives et al proposed the noise-aware physics-informed machine learning framework [33] to train PINN based on discrete Fourier transform in a multitasking learning paradigm [34] to reveal a set of optimal reduced partial differential equations. PDE-Read [35] introduces a variant of the two-network and a Recursive Feature Elimination algorithm to identify human readable PDEs from data.…”

Section: Introductionmentioning

confidence: 99%

Governing equation discovery based on causal graph for nonlinear dynamic systems

Jia,

Zhou,

et al. 2023

The governing equations of nonlinear dynamic systems is of great significance for understanding the internal physical characteristics. In order to learn the governing equations of nonlinear systems from noisy observed data, we propose a novel method named governing equation discovery based on causal graph that combines spatio-temporal graph convolution network with governing equation modeling. The essence of our method is to first devise the causal graph encoding based on transfer entropy to obtain the adjacency matrix with causal significance between variables. Then, the spatio-temporal graph convolutional network is used to obtain approximate solutions for the system variables. On this basis, automatic differentiation is applied to obtain basic derivatives and form a dictionary of candidate algebraic terms. Finally, sparse regression is used to obtain the coefficient matrix and determine the explicit formulation of the governing equations. We also design a novel cross-combinatorial optimization strategy to learn the heterogeneous parameters that include neural network parameters and control equation coefficients. We conduct extensive experiments on seven datasets from different physical fields. The experimental results demonstrate the proposed method can automatically discover the underlying governing equation of the systems, and has great robustness.