DEEP NURBSADMISSIBLE PHYSICS-INFORMED NEURAL NETWORKS HAMED SAIDAOUI LUIS ESPATH1 RA UL TEMPONE234 Abstract. In this study we propose a new numerical scheme for physics-informed neural networks PINNs

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DEEP NURBS—ADMISSIBLE PHYSICS-INFORMED NEURAL NETWORKS

HAMED SAIDAOUI♯, LUIS ESPATH1& RA ´

UL TEMPONE2,3,4

Abstract. In this study, we propose a new numerical scheme for physics-informed neural networks (PINNs)

that enables precise and inexpensive solutions for partial diﬀerential equations (PDEs) in case of arbitrary

geometries while strongly enforcing Dirichlet boundary conditions. The proposed approach combines ad-

missible NURBS parametrizations (admissible in the calculus of variations sense, that is, satisfying the

boundary conditions) required to deﬁne the physical domain and the Dirichlet boundary conditions with

a PINN solver. Therefore, the boundary conditions are automatically satisﬁed in this novel Deep NURBS

framework. Furthermore, our sampling is carried out in the parametric space and mapped to the physical

domain. This parametric sampling works as an importance sampling scheme since there is a concentration of

points in regions where the geometry is more complex. We veriﬁed our new approach using two-dimensional

elliptic PDEs when considering arbitrary geometries, including non-Lipschitz domains. Compared to the

classical PINN solver, the Deep NURBS estimator has a remarkably high accuracy for all the studied prob-

lems. Moreover, a desirable accuracy was obtained for most of the studied PDEs using only one hidden

layer of neural networks. This novel approach is considered to pave the way for more eﬀective solutions for

high-dimensional problems by allowing for a more realistic physics-informed statistical learning framework

to solve PDEs.

AMS subject classiﬁcations: ·35L65 ·

Contents

1. Introduction 1

2. Mathematical background 3

2.1. Construction of admissible NURBS parameterizations 4

3. Deep NURBS 5

3.1. Partial diﬀerential equation and method overview 6

3.2. Physics - Informed Neural Networks (PINNs) 7

4. Numerical results 10

4.1. Physical Domain with Corner Singularity 10

4.2. Annulus geometry 13

4.3. Square with a hole 14

5. Conclusion 16

6. Declarations: Funding and/or Conﬂicts of interests/Competing interests & Data availability 17

7. Additional Declaration 17

References 17

1. Introduction

Deep Learning (DL) exhibits unprecedented advancement in the last two decades [1]. Its usage in numerous

disciplines has resulted in diverse successful implementations such as language processing [2, 3] and image

1School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom

2Department of Mathematics, RWTH Aachen University, Geb¨

aude-1953 1.OG, Pontdriesch 14-16, 161, 52062

Aachen, Germany.

3King Abdullah University of Science & Technology (KAUST), Computer, Electrical and Mathematical Sci-

ences & Engineering Division (CEMSE), Thuwal 23955-6900, Saudi Arabia.

4Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen Univer-

sity, Germany.

E-mail address:♯hamed.saidaoui@kaust.edu.sa.

Date: July 30, 2024.

arXiv:2210.13900v2 [math.NA] 29 Jul 2024

2 DEEP NURBS—ADMISSIBLE PHYSICS-INFORMED NEURAL NETWORKS

recognition [4, 5]. With the exception of a few studies, the application of machine learning (ML) along

with neural networks (NNs) in general, in particular to scientiﬁc problems, was not as structured as in the

previous topics (for example [6, 7, 8, 9, 10, 11]).

Among these few eﬀorts was the seminal work of Lagaris [6], who applied very simplistic NN models

to solve several ordinary and partial diﬀerential equations (PDEs) of diﬀerent orders (the accuracy was

order-dependent though). This concept has been revived very recently by many researchers [12, 13, 14, 15]

who used a very similar concept and coined it as ”physics-informed neural networks (PINNs)” (two such

examples were deep Ritz and deep Galerkin for the last two references, respectively). Since then, PINNs

have gained considerable fame because of their simple implementation and their concept that allows for a

combination of NNs and the already established theories. PINNs have enabled the shift to the unsupervised

ML scheme owing to the laws and constraints implemented within its framework. Targeting the solution of

PDEs, PINNs have been proven to be very eﬃcient for many complicated and challenging problems ([16, 17])

and for diﬀerent types of PDEs. Furthermore, the convenience of the accompanying sampling methods made

it adequate for problems with high-dimensional domains. Because PINNs have been speciﬁcally proposed

for methods that consider the collocation points as training data with a discrete loss term; note that other

variants relying on the same principle coexisted, e.g., deep Ritz [14, 18] and deep Galerkin [15]. The ﬁrst

method uses the weak form of the PDE in which the variational problem (energy minimization) can be

solved using stochastic optimizers and Monte Carlo sampling techniques. In the deep Galerkin scheme, the

gradients have been taken care of using a tailored stochastic approach.

Many variations have been proposed and proven to be eﬀective in speciﬁc cases, in addition to these

three primary methods. Inverse problems have been tackled using stochastic diﬀerential equations (SDEs)

in this reference [19]. Lu et al. [20] proposed a method for learning non-linear operators, while Cai et

al. [16] have implemented a modiﬁed version of PINNs to infer electroconvection in multiphysics systems.

A comprehensive review on ML for ﬂuid mechanics is presented in [21]. The mathematical formulation of

PINNs has been discussed in certain relevant studies from an uncertainty quantiﬁcation perspective. Mishra

et al. [22] provided an estimate for its generalization error. They primarily provided bounds for PINNs

based on quadrature and random sampling based-PINNs. Fang et al. [23] established convergence rates for

the NNs-based PDE solver. To develop an approximation for the derivatives, they combined the NNs and

the diﬀerential operator (as it is used in Finite element methods).

As a trend in applying ML tools to scientiﬁc and engineering problems, we naturally wonder what makes

PINNs and their variants (deep Ritz and deep Galerkin) as eﬃcient and vulnerable as they are (or considered

to be). To address this question, one should examine what makes these methods diﬀerent from answering

this question. Surely its simple concept makes it easy to implement, particularly for complicated tasks in

which simplicity plays an important role in its success. One has to admit that the nature of its loss function,

which incorporates the system’s physical laws, has to do with its convergence rate to the sought solution.

The auto-diﬀerentiation, being an exact, diﬀerentiating tool, is, in turn, at the heart of PINNs success.

However, PINNs have observed many limitations when it comes to problems with discontinuities and

computationally demanding problems [14]. These problems are very demanding in terms of NN depth and

computational work; the latter two aspects make the PINNs usage challenging. From this perspective, one

might question what has to change to allow PINNs to cope with discontinuities without being extremely

expensive. Is it the way the loss function is presented (weak form [14, 18] or strong form [12]), or is it related

to the way the gradients are calculated (entirely auto-diﬀerentiation - based or hybrid [23]). If we examine

the previous references, one can conclude that none of the previous aspects is important in mitigating the

complexity of the NN architecture.

Shin et al. [24] demonstrated the sequence of minimizers generated in the stochastic optimization of

PINNs strongly converges to the solution of the PDE in C0. However, if the boundary conditions are satis-

ﬁed, the minimizers converge to the sought solution in H1instead of C0. This insight has been supported

by recent works [25, 13] and genuinely by the original work of Lagaris [6]. The Lagaris approach was to

use functions that naturally fulﬁll the boundary conditions. Although this approach proved to be compu-

tationally very eﬃcient, in terms of accuracy, this approach is not practical regarding complex geometries

with non-homogeneous boundaries. From this last point, we follow the concept described in Lagaris’ paper.

Nevertheless, we use admissible (in the sense of calculus of variations, that is, satisfying the boundary condi-

tions) non-uniform rational B-splines (NURBS) parameterizations, which are similarly used in IsoGeometric

Analysis, as an eﬃcient approach to impose Dirichlet boundary conditions. To our knowledge, our work

DEEP NURBS—ADMISSIBLE PHYSICS-INFORMED NEURAL NETWORKS 3

(a) Admissible

(ansatz) NURBS φ(x)

(b) Neural Network

NN(x)

Network N N (x)φ(x)

Figure 1. Exempliﬁcation of Deep NURBS’ method. From left to right: Admissible

NURBS (ansatz) φ(x), Neural Network NN(x), and the product of the two ﬁrst, that

is, Admissible Neural Network NN(x)φ(x), for the homogeneous Dirichlet case.

is the ﬁrst to impose boundary conditions using admissible NURBS in the PINNs framework. We call our

method Deep-NURBS, which is a combination of deep neural networks (DNNs) and NURBS. We use the

weak form of the loss function (energy term), and we implement our algorithms in Tensorﬂow [26]. Figure 1

shows the admissible NURBS (which works as an ansatz), the NN, and the product of the two, rendering the

Admissible NN that ultimately leads to the Deep-NURBS method. Moreover, despite being an ML-based

method, we demonstrate that our method is robust against slight changes in the NN parameters. This fact

allows us to overcome computational costs coming with hyperparameter optimization. Lastly, note that the

NURBS parameterization does not change the overall computational cost, that is, our Deep NURBS method

increases the approximability of the Neural Networks without increasing the overall cost for a ﬁx Neural

Network.

In this study, we overview NURBS parameterizations in Section 2. In Section 3, we discuss our approach

based on combining DNNs with NURBS. Section 4 will be devoted to discussing our result of applying deep

NURBS to various problems with non-trivial complex geometries. Finally, we wrap up this study with a

conclusion.

2. Mathematical background

Our notation is as follows: scalar and vectorial ﬁelds are denoted by lower-case letters and lower-case

boldface letters, respectively. The gradient and Laplacian operators are denoted by ∇and △, respectively.

The physical domain is denoted as Dwith boundary ∂D. Moreover, we let ∂D:=∂DD∪∂DNwhere the

Dirichlet boundary conditions are considered on DDwhile the Neumann type of boundary conditions are

considered on DN. We construct our approximation on the Sobolev space of all square-integrable functions

on Dwith square-integrable derivatives. Thus, we denote the Lebesgue space as L2and L2for scalar-valued

and vector-valued functions, respectively. Similarly, we denote the Sobolev space of scalar- and vector-valued

functions as Hkand Hk, respectively, where the norm of the kth gradient is L2.

Let us consider two cases. Let L[u(x;θ)] be (a) the pointwise residual of a PDE or (b) the energy density

of the underlying system given an ansatz uparametrized by θ∈RQwhere Qis the total number of degrees

of freedom (e.g., the total number of neural network parameters). Next, let u∗be the solution of this PDE

with values uD, on ∂DDwhile its normal derivative ∂nu∗takes the value gNon ∂DN. In a learning type of

framework, we may state problems (a) and (b) as follows,

(1) θ∗:= arg min

θ∈RQ∥L[u(·;θ)]∥2

L2(D)+ Dirichlet terms on ∂DD+ Neumann terms on ∂DN.

Note that the Dirichlet term is imposed in a weak sense where authors in Ref. [14] use a penalization term

to impose the Dirichlet boundary conditions. Instead, our approach is to impose the Dirichlet boundary

conditions in a strong sense. To this end, we consider a parameterization of our domain D,

(2) x:=χ(ξ),

4 DEEP NURBS—ADMISSIBLE PHYSICS-INFORMED NEURAL NETWORKS

where we assume that χis a bijective mapping with x∈ D and ξis a parameter that lives in the parameter

space Ξ. We here use the notion of admissible ﬁelds. A family of functions is said to be admissible

if Dirichlet boundary conditions are satisﬁed. Consider the following family of admissible vector-valued

parameterizations

(3) Z:={ζ|ζ∈Hk(D)∧ζ(x) = uD(x),∀x=χ(ξ)∈∂DD}.

Moreover, we deﬁne a family of auxiliary smooth scalar-valued functions Pwith vanishing boundary, i.e.,

φ:χ(Ξ)7→ Rwith φ(χ(ξ)) = 0 for all ξ=χ−1(x) such that x=χ(ξ)∈∂DD. Furthermore, we require

φ∈Hk(D). Thus,

(4) P:={φ|φ∈Hk(D)∧φ(x)=0,∀x=χ(ξ)∈∂DD}.

The problem statement (1) reads

(5) 





θ∗:= arg min

θ∈RQ∥L[φ(·)u(·;θ) + ζ(·)]∥2

L2(D)+ Neumann terms on ∂DN,

given an arbitrary φ∈ P and ζ∈ Z.

Furthermore, for the scalar case, that is, we replace the vector-valued function uwith the scalar valued

function u, expression (5) takes the following form

(6) 





θ∗:= arg min

θ∈RQ∥L[φ(·)u(·;θ) + ζ(·)])∥2

L2(D)+ Neumann terms on ∂DN,

given an arbitrary φ∈ P and ζ∈ Z,

where here Zis

(7) Z:={ζ|ζ∈Hk(D)∧ζ(x) = uD(x),∀x=χ(ξ)∈∂DD}.

If we instead restrict attention to the scalar case with homogeneous boundary conditions, the function φ

becomes our admissible ﬁeld such that the expression (6) specializes to

(8) 





θ∗:= arg min

θ∈RQ∥L[φ(·)u(·;θ)])∥2

L2(D)+ Neumann terms on ∂DN,

given an arbitrary φ∈ P.

We will demonstrate how φand ζmay be developed using NURBS parameterizations.

2.1. Construction of admissible NURBS parameterizations. Now, let us consider the subset of

NURBS functions Pλ⊂ P and Zλ⊂ Z which will work as an ansatz. The Cox–deBoor recursive for-

mulation [27, 28] is usually adopted to evaluate B-spline basis functions based on [29]. In ddimensions,

B-splines are obtained considering a given knot vector Ξ=Nd

ȷ=1 Ξȷdeﬁned over the parametric space with

polynomial degree pȷalong the parametric direction ξȷ. Finally, to map from the parametric space to the

physical one, we use nȷ+ 1 control points. The NURBS basis functions and physical domain are discussed

below:

Bases: B-splines in the ȷth spatial direction

Ni,0(ξ) = (1 if ξi⩽ξ < ξi+1,

0 otherwise.

Ni,p (ξ) = ξ−ξi

ξi+p−ξi

Ni,p−1(ξ) + ξi+p+1 −ξ

ξi+p+1 −ξi+1

Ni+1,p−1(ξ)

(9)

with

(10) Ξȷ={0,...,0

| {z }

pȷ+1

, ξȷ

pȷ+1, . . . , ξȷ

sȷ−pȷ−1,1,...,1

| {z }

pȷ+1 },

where sȷ=nȷ+pȷ+ 1. nȷis the number of basis functions and pȷits degree along direction ȷ.

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DEEPNURBS—ADMISSIBLEPHYSICS-INFORMEDNEURALNETWORKSHAMEDSAIDAOUI♯,LUISESPATH1&RA´ULTEMPONE2,3,4Abstract.Inthisstudy,weproposeanewnumericalschemeforphysics-informedneuralnetworks(PINNs)thatenablespreciseandinexpensivesolutionsforpartialdifferentialequations(PDEs)incaseofarbitrarygeometrieswhilestrongl...

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