Systematic Construction of Neural Forms for Solving Partial Differential Equations Inside Rectangular Domains, Subject to Initial, Boundary and Interface Conditions

Lagari, Pola Lydia; Tsoukalas, Lefteri H.; Safarkhani, Salar; Lagaris, Isaac E.

doi:10.1142/s0218213020500098

Cited by 40 publications

(21 citation statements)

References 11 publications

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“…If \Gamma D is a simple geometry, then it is possible to choose \ell (x) analytically [31,32,42]. For example, when \Gamma D is the boundary of an interval \Omega = [a, b], i.e., \Gamma D = \{ a, b\} , we can choose \ell (x) as (xa)(bx) or (1e a - x )(1e x - b ).…”

Section: 3mentioning

confidence: 99%

Physics-Informed Neural Networks with Hard Constraints for Inverse Design

Lu¹,

Pestourie²,

Yao³

et al. 2021

SIAM J. Sci. Comput.

284

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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Inverse design arises in a variety of areas in engineering such as acoustic, mechanics, thermal/electronic transport, electromagnetism, and optics. Topology optimization is an important form of inverse design, where one optimizes a designed geometry to achieve targeted properties parameterized by the materials at every point in a design region. This optimization is challenging, because it has a very high dimensionality and is usually constrained by partial differential equations (PDEs) and additional inequalities. Here, we propose a new deep learning method---physics-informed neural networks with hard constraints (hPINNs)---for solving topology optimization. hPINN leverages the recent development of PINNs for solving PDEs, and thus does not require a large dataset (generated by numerical PDE solvers) for training. However, all the constraints in PINNs are soft constraints, and hence we impose hard constraints by using the penalty method and the augmented Lagrangian method. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow. We achieve the same objective as conventional PDE-constrained optimization methods based on adjoint methods and numerical PDE solvers, but find that the design obtained from hPINN is often smoother for problems whose solution is not unique. Moreover, the implementation of inverse design with hPINN can be easier than that of conventional methods because it exploits the extensive deep-learning software infrastructure.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . inverse design, topology optimization, partial differential equations, physicsinformed neural networks, penalty method, augmented Lagrangian method \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 35R30, 65K10, 68T20 \bfD \bfO \bfI .

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Section: 3mentioning

confidence: 99%

Physics-Informed Neural Networks with Hard Constraints for Inverse Design

Lu¹,

Pestourie²,

Yao³

et al. 2021

SIAM J. Sci. Comput.

284

View full text Add to dashboard Cite

show abstract

“…The trial solution (TS) method has been proposed for the approximation of a given differential equation, which we abbreviate here as TSM [9]. The TS, also called neural form in [9,10], has to contain the neural network output and has to satisfy given initial or boundary conditions by construction. Under the latter conditions, there are multiple different possible forms of the TS for the approximation of a differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…Under the latter conditions, there are multiple different possible forms of the TS for the approximation of a differential equation. Recently, a systematic construction approach for the TS has been proposed [10]. However, as indicated in the latter work, the TS construction may become difficult to realise for complex problems.…”

Section: Introductionmentioning

confidence: 99%

“…However, one may wonder about the direct comparison of TSM and mTSM in terms of quality of results as well as in the related computational aspects. Let us stress in this context, that the original works [9,10,12] mainly describe the network architecture and elaborate on the TS and mTSM construction, but they do not contain more details of the computational characteristics of the methods. Yet it turns out that it is not trivial to define a computational framework that gives competitive results.…”

Section: Introductionmentioning

confidence: 99%

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Computational characteristics of feedforward neural networks for solving a stiff differential equation

Schneidereit

Breuß

2022

Neural Comput & Applic

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Feedforward neural networks offer a possible approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature. Computational approaches are in general highly dependent on a variety of computational parameters as well as on the choice of optimisation methods, a point that has to be seen together with the structure of the cost function. The intention of this paper is to make a step towards resolving these open issues. To this end, we study here the solution of a simple but fundamental stiff ordinary differential equation modelling a damped system. We consider two computational approaches for solving differential equations by neural forms. These are the classic but still actual method of trial solutions defining the cost function, and a recent direct construction of the cost function related to the trial solution method. Let us note that the settings we study can easily be applied more generally, including solution of partial differential equations. By a very detailed computational study, we show that it is possible to identify preferable choices to be made for parameters and methods. We also illuminate some interesting effects that are observable in the neural network simulations. Overall we extend the current literature in the field by showing what can be done in order to obtain useful and accurate results by the neural network approach. By doing this we illustrate the importance of a careful choice of the computational setup.

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“…The cost function in this context has the characteristic of connecting the DE with the neural network framework [5,6]. This may be achieved with so-called neural forms (NFs), which are in fact trial solutions satisfying the given conditions [7,8,9]. The neural forms approach results in an unsupervised learning framework.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Neural Domain Refinement for Solving Time-Dependent Differential Equations

Schneidereit¹,

Breuß²

2021

Preprint

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A classic approach for solving differential equations with neural networks builds upon neural forms, in which a cost function can be constructed directly using the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent differential equations, one can apply the recently developed method of domain fragmentation. That is, the domain may be split into several subdomains, on which the optimisation problem is solved.In classic adaptive numerical methods for solving differential equations, the mesh as well as the domain may be refined or decomposed, respectively, in order to improve accuracy. Also the degree of approximation accuracy may be adapted. It would be desirable to transfer such important and successful strategies to the field of neural network based solutions. In the present work, we propose a novel adaptive neural approach to meet this aim for solving time-dependent problems.To this end, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks employed are by default small, the number of neurons may also be adjusted in an adaptive way. We introduce conditions to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary. We provide results for three carefully chosen example initial value problems and illustrate important properties of the method alongside.

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Systematic Construction of Neural Forms for Solving Partial Differential Equations Inside Rectangular Domains, Subject to Initial, Boundary and Interface Conditions

Cited by 40 publications

References 11 publications

Physics-Informed Neural Networks with Hard Constraints for Inverse Design

Physics-Informed Neural Networks with Hard Constraints for Inverse Design

Computational characteristics of feedforward neural networks for solving a stiff differential equation

Adaptive Neural Domain Refinement for Solving Time-Dependent Differential Equations

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