PDE-READ: Human-readable partial differential equation discovery using deep learning

Stephany, Robert; Earls, Christopher J.

doi:10.1016/j.neunet.2022.07.008

Cited by 17 publications

(3 citation statements)

References 14 publications

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“…The governing equation for this example constitutes a fourth-order spatial partial derivative and a second-order temporal partial derivative. The higher-order spatial and temporal derivatives in the PDEs amplify noise in the training process, which makes identifying the unknown parameter in this example a challenging task [ 53 - 54 ] .…”

Section: Resultsmentioning

confidence: 99%

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Daneker

Jolley

et al. 2023

Appl. Math. Mech.-Engl. Ed.

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Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.

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Section: Resultsmentioning

confidence: 99%

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Daneker

Jolley

et al. 2023

Appl. Math. Mech.-Engl. Ed.

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“…However, we note that most of the differential equations in physics may be expanded to this form. There are some existing algorithms based on neural networks to find equations in a more compact form [43]. However, they require a priori knowledge of the equation form.…”

Section: Evolutionary Optimization Algorithmmentioning

confidence: 99%

Methods of Partial Differential Equation Discovery: Application to Experimental Data on Heat Transfer Problem

Andreeva,

Bykov,

Gataulin

et al. 2023

Processes

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The paper presents two effective methods for discovering process models in the form of partial differential equations based on an evolutionary algorithm and an algorithm for the best subset selection. The methods are designed to work with sparse and noisy data and implement various numerical differentiation techniques, including piecewise local approximation using multidimensional polynomial functions, neural network approximation, and an additional algorithm for selecting differentiation steps. To verify the algorithms, the experiment is carried out on pulsed heating of a viscous liquid (glycerol) by a submerged horizontal cylindrical heat source. Temperature measurements are taken only at six points, which makes the data very sparse. The noise level ranges from 0.2 to 1% of the observed maximum temperature. The algorithms can successfully restore the structure of the heat transfer equation in cylindrical coordinates and determine the thermal diffusivity coefficient with an error of 2.5–20%, depending on the algorithm type and heating mode. Additional synthetic setups are employed to analyze the dependence of accuracy on the noise level. Results also demonstrate the algorithms’ ability to identify underlying processes such as convective motion.

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“…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. Chen et al proposed alternating direction optimization by combining PINN with STRidge to successfully identify differential equation coefficients [28].…”

Section: Introductionmentioning

confidence: 99%

Governing equation discovery based on causal graph for nonlinear dynamic systems

Jia,

Zhou,

et al. 2023

Mach. Learn.: Sci. Technol.

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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.

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PDE-READ: Human-readable partial differential equation discovery using deep learning

Cited by 17 publications

References 14 publications

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Methods of Partial Differential Equation Discovery: Application to Experimental Data on Heat Transfer Problem

Governing equation discovery based on causal graph for nonlinear dynamic systems

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