Solving Partial Differential Equations Using Deep Learning and Physical Constraints

Guo, Yanan; Cao, Xiaoqun; Liu, Bainian; Gao, Mei

doi:10.3390/app10175917

Cited by 74 publications

(46 citation statements)

References 57 publications

(63 reference statements)

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“…As surrogate E-fields are not required by our method, realistic head models from diverse imaging datasets can be used as the training data for our method. To the best of our knowledge, our method is the first study to investigate self-supervised deep learning for TMS E-field modeling, though physics-informed deep learning methods have been successfully applied to solving PDEs on varied domains [34][35][36][37][38][39][40][41][42].…”

Section: Discussionmentioning

confidence: 99%

“…To train the DNNs, E-fields estimated by conventional numerical methods, such as FEM, are used to generate training data. Therefore, their accuracy would be bounded by the conventional numerical methods used Inspired by self-supervised deep learning methods [34][35][36][37][38][39][40][41][42][43][44][45] and the pioneer deep learning based E-field computation methods [32,33], we develop a novel self-supervised deep learning based TMS Efield modeling method to obtain precise high-resolution E-fields in real-time. Specially, given a head model and the primary E-field generated by TMS coil, a DL model is built to generate the electric scalar potential by minimizing a loss function that measures how well the generated electric scalar potential fits the governing PDE and Neumann boundary condition, from which the E-field can be derived directly.…”

Section: Introductionmentioning

confidence: 99%

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Computation of transcranial magnetic stimulation electric fields using self-supervised deep learning

Li¹,

Deng

Oathes

et al. 2021

Preprint

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BackgroundElectric fields (E-fields) induced by transcranial magnetic stimulation (TMS) can be modeled using partial differential equations (PDEs) with boundary conditions. However, existing numerical methods to solve PDEs for computing E-fields are usually computationally expensive. It often takes minutes to compute a high-resolution E-field using state-of-the-art finite-element methods (FEM).MethodsWe developed a self-supervised deep learning (DL) method to compute precise TMS E-fields in real-time. Given a head model and the primary E-field generated by TMS coils, a self-supervised DL model was built to generate a E-field by minimizing a loss function that measures how well the generated E-field fits the governing PDE and Neumann boundary condition. The DL model was trained in a self-supervised manner, which does not require any external supervision. We evaluated the DL model using both a simulated sphere head model and realistic head models of 125 individuals and compared the accuracy and computational efficiency of the DL model with a state-of-the-art FEM.ResultsIn realistic head models, the DL model obtained accurate E-fields with significantly smaller PDE residual and boundary condition residual than the FEM (p<0.002, Wilcoxon signed-rank test). The DL model was computationally efficient, which took about 0.30 seconds on average to compute the E-field for one testing individual. The DL model built for the simulated sphere head model also obtained an accurate E-field whose difference from the analytical E-fields was 0.004, more accurate than the solution obtained using the FEM.ConclusionsWe have developed a self-supervised DL model to directly learn a mapping from the magnetic vector potential of a TMS coil and a realistic head model to the TMS induced E-fields, facilitating real-time, precise TMS E-field modeling.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computation of transcranial magnetic stimulation electric fields using self-supervised deep learning

Li¹,

Deng

Oathes

et al. 2021

Preprint

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“…[26] used PINN to solve the nonlinear Eikonal equation and demonstrated the transfer learning possibilities of PINN. Guo et al [27] verified PINN's ability in solving the wave equation, the KdV-Burgers equation, and the KdV equation. In addition, PINN was also applied to the uncertainty quantification problem [28,29] and the atomic simulation of materials [30].…”

Section: Introductionmentioning

confidence: 99%

A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN)

Yan

et al. 2022

Algorithms

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The problem of electro-thermal coupling is widely present in the integrated circuit (IC). The accuracy and efficiency of traditional solution methods, such as the finite element method (FEM), are tightly related to the quality and density of mesh construction. Recently, PINN (physics-informed neural network) was proposed as a method for solving differential equations. This method is mesh free and generalizes the process of solving PDEs regardless of the equations’ structure. Therefore, an experiment is conducted to explore the feasibility of PINN in solving electro-thermal coupling problems, which include the electrokinetic field and steady-state thermal field. We utilize two neural networks in the form of sequential training to approximate the electric field and the thermal field, respectively. The experimental results show that PINN provides good accuracy in solving electro-thermal coupling problems.

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“…al. [19]. This improved PINN takes the physical information in PDE as a regularization term, which improves the performance of neural networks.…”

Section: Introductionmentioning

confidence: 99%

ANNs-Based Method for Solving Partial Differential Equations : A Survey

Pratama¹,

Bakar²,

Mohan³

et al. 2021

Preprint

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Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.

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Solving Partial Differential Equations Using Deep Learning and Physical Constraints

Cited by 74 publications

References 57 publications

Computation of transcranial magnetic stimulation electric fields using self-supervised deep learning

Computation of transcranial magnetic stimulation electric fields using self-supervised deep learning

A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN)

ANNs-Based Method for Solving Partial Differential Equations : A Survey

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