A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks Suchuan Dong Yiran Wang

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A Method for Computing Inverse Parametric PDE Problems with

Random-Weight Neural Networks

Suchuan Dong∗

, Yiran Wang

Center for Computational and Applied Mathematics

Department of Mathematics

Purdue University, USA

(October 8, 2022)

Abstract

We present a method for computing the inverse parameters and the solution ﬁeld to inverse parametric

partial diﬀerential equations (PDE) based on randomized neural networks. This extends the local extreme

learning machine technique originally developed for forward PDEs to inverse problems. We develop three

algorithms for training the neural network to solve the inverse PDE problem. The ﬁrst algorithm (termed

NLLSQ) determines the inverse parameters and the trainable network parameters all together by the

nonlinear least squares method with perturbations (NLLSQ-perturb). The second algorithm (termed

VarPro-F1) eliminates the inverse parameters from the overall problem by variable projection to attain

a reduced problem about the trainable network parameters only. It solves the reduced problem ﬁrst

by the NLLSQ-perturb algorithm for the trainable network parameters, and then computes the inverse

parameters by the linear least squares method. The third algorithm (termed VarPro-F2) eliminates the

trainable network parameters from the overall problem by variable projection to attain a reduced problem

about the inverse parameters only. It solves the reduced problem for the inverse parameters ﬁrst, and

then computes the trainable network parameters afterwards. VarPro-F1 and VarPro-F2 are reciprocal

to each other in some sense. The presented method produces accurate results for inverse PDE problems,

as shown by the numerical examples herein. For noise-free data, the errors of the inverse parameters and

the solution ﬁeld decrease exponentially as the number of collocation points or the number of trainable

network parameters increases, and can reach a level close to the machine accuracy. For noisy data, the

accuracy degrades compared with the case of noise-free data, but the method remains quite accurate.

The presented method has been compared with the physics-informed neural network method.

Keywords: randomized neural networks, extreme learning machine, nonlinear least squares, variable pro-

jection, inverse problems, inverse PDE

1 Introduction

In this work we focus on the simultaneous determination of the parameters (as constants or ﬁeld distributions)

and the solution ﬁeld to parametric PDEs based on artiﬁcial neural networks (ANN/NN), given sparse and

noisy measurement data of certain variables. This type of problems is often referred to as the inverse PDE

problems in the literature [31]. Typical examples include the determination of the diﬀusion coeﬃcient given

certain concentration data or the computation of the wave speed given sparse measurement of the wave

proﬁle. When the parameter values in the PDE are known, approximation of the PDE solution is often

referred to as the forward PDE problem. We will adopt these notations in this paper.

Closely related to the inverse PDE problems is the data-driven “discovery” of PDEs (see e.g. [4, 52,

7, 48, 49, 44, 60, 2, 34, 47, 54, 5, 55, 40], among others), in which, given certain measurement data, the

functional form of the PDE is to be discerned. In order to acquire a parsimonious PDE form, techniques

such as sparsity promotion [7, 48] or dimensional analysis [60] are often adopted.

∗Author of correspondence. Emails: sdong@purdue.edu (S. Dong), wang2335@purdue.edu (Y. Wang).

arXiv:2210.04338v1 [math.NA] 9 Oct 2022

As advocated in [53, 31], data-driven scientiﬁc machine learning problems can be viewed in terms of the

amount of data that is available and the amount of physics that is known. They are broadly classiﬁed into

three categories in [31]: (i) those with “lots of physics and small data” (e.g. forward PDE problems), (ii)

those with “some physics and some data” (e.g. inverse PDE problems), and (iii) those with “no physics and

big data” (e.g. general PDE discovery). The authors of [31] point out that those in the second category are

typically the more interesting and representative in real applications, where the physics is partially known

and sparse measurements are available. One illustrating example is from multiphase ﬂows, where the conser-

vation laws (mass/momentum conservations) and thermodynamic principles (second law of thermodynamics,

Galilean invariance) lead to a thermodynamically-consistent phase ﬁeld model, but with an incomplete sys-

tem of governing equations [15, 14]. One has the freedom to choose the form of the free energy, the wall

energy, the form and coeﬃcients of the constitutive relation, and the form and coeﬃcient of the interfacial

mobility [12, 13, 58]. Diﬀerent choices will lead to diﬀerent speciﬁc models, which are all thermodynami-

cally consistent. The diﬀerent models cannot be distinguished by the thermodynamic principles, but can be

diﬀerentiated with experimental measurements.

The development of machine learning techniques for solving inverse PDE problems has attracted a great

deal of interest recently, with a variety of contributions from diﬀerent researchers. In [44] a method for

estimating the parameters in nonlinear PDEs is developed based on Gaussian processes, where the state

variable at two consecutive snapshots are assumed to be known. The physics informed neural network

(PINN) method is introduced in the inﬂuential work [45] for solving forward and inverse nonlinear PDEs.

The residuals of the PDE, the boundary and initial conditions, and the measurement data are encoded

into the loss function as soft constraints, and the neural network is trained by gradient descent or back

propagation type algorithms. The PINN method has inﬂuenced signiﬁcantly subsequent developments and

stimulated applications in many related areas (see e.g. [36, 46, 29, 38, 11, 50, 9, 35, 56, 30, 43], among

others). A hybrid ﬁnite element and neural network method is developed in [1]. The ﬁnite element method

(FEM) is used to solve the underlying PDE, which is augmented by a neural network for representing the

PDE coeﬃcient [1]. A conservative PINN method is proposed in [29] together with domain decomposition

for simulating nonlinear conservation laws, in which the ﬂux continuity is enforced along the sub-domain

interfaces, and interesting results are presented for a number of forward and inverse problems. This method

is further developed and extended in a subsequent work [28] with domain decompositions in both space and

time; see a recent study in [30] of this extended technique for supersonic ﬂows. Interesting applications

are described in [46, 9], where the PINN technique is employed to infer the 3D velocity and pressure ﬁelds

based on scattered ﬂow visualization data or Schlieren images from experiments. In [20] a distributed PINN

method based on domain decomposition is presented, and the loss function is optimized by a gradient descent

algorithm. For nonlinear PDEs, the method solves a related linearized equation with certain variables ﬁxed

at their initial values [20]. An auxiliary PINN technique is developed in [59] for solving nonlinear integro-

diﬀerential equations, in which auxiliary variables are introduced to represent the anti-derivatives and thus

avoiding the integral computation. We would also like to refer the reader to e.g. [11, 53, 37, 33] (among

others) for inverse applications of neural networks in other related ﬁelds.

In the current work we consider the use of randomized neural networks, also known as extreme learning

machines (ELM) [25] (or random vector functional link (RVFL) networks [42]), for solving inverse PDE

problems. ELM was originally developed for linear classiﬁcation and regression problems. It is characterized

by two ideas: (i) randomly assigned but ﬁxed (non-trainable) hidden-layer coeﬃcients, and (ii) trainable

linear output-layer coeﬃcients determined by linear least squares or by using the Moore-Penrose inverse [25].

This technique has been extended to scientiﬁc computing in the past few years, for function approximations

and for solving ordinary and partial diﬀerential equations (ODE/PDE); see e.g. [57, 41, 21, 16, 17, 10, 22, 51,

19], among others. The random-weight neural networks are universal function approximators. As established

by the theoretical results of [27, 26, 39], a single-hidden-layer feed-forward neural network (FNN) having

random but ﬁxed (not trained) hidden units can approximate any continuous function to any desired degree

of accuracy, provided that the number of hidden units is suﬃciently large.

In this paper we present a method for computing inverse PDE problems based on randomized neural

networks. This extends the local extreme learning machine (locELM) technique originally developed in [16]

for forward PDEs to inverse problems. Because of the coupling between the unknown PDE parameters

(referred to as the inverse parameters hereafter) and the solution ﬁeld, the inverse PDE problem is fully

nonlinear with respect to the unknowns, even though the associated forward PDE may be linear. We

partition the overall domain into sub-domains, and represent the solution ﬁeld (and the inverse parameters,

if they are ﬁeld distributions) by a local FNN on each sub-domain, imposing Ck(with appropriate k)

continuity conditions across the sub-domain boundaries. The weights/biases in the hidden layers of the

local NNs are assigned to random values and ﬁxed (not trainable), and only the output-layer coeﬃcients are

trainable. The inverse PDE problem is thus reduced to a nonlinear problem about the inverse parameters

and the output-layer coeﬃcients of the solution ﬁeld, or if the inverse parameters are ﬁeld distributions,

about the output-layer coeﬃcients for the inverse parameters and the solution ﬁeld.

We develop three algorithms for training the neural network to solve the inverse PDE problem:

•The ﬁrst algorithm (termed NLLSQ) computes the inverse parameters and the trainable parameters

of the local NNs all together by the nonlinear least squares method [3]. This extends the nonlinear

least squares method with perturbations (NLLSQ-perturb) from [16] (developed for forward nonlinear

PDEs) to inverse PDE problems.

•The second algorithm (termed VarPro-F1) eliminates the inverse parameters from the overall problem

based on the variable projection (VarPro) strategy [23, 24] to attain a reduced problem about the

trainable network parameters only. It solves the reduced problem ﬁrst for the trainable parameters

of the local NNs by the NLLSQ-perturb algorithm, and then computes the inverse parameters by the

linear least squares method.

•The third algorithm (termed VarPro-F2) eliminates the trainable network parameters from the overall

inverse problem by variable projection to arrive at a reduced problem about the inverse parameters

only. It solves the reduced problem ﬁrst for the inverse parameters by the NLLSQ-perturb algorithm,

and then computes the trainable parameters of the local NNs based on the inverse parameters already

obtained. The VarPro-F2 and VarPro-F1 algorithms both employ the variable projection idea and are

reciprocal formulations in a sense. For inverse problems with an associated forward nonlinear PDE,

VarPro-F2 needs to be combined with a Newton iteration.

The presented method produces accurate solutions to inverse PDE problems, as shown by a number of

numerical examples presented herein. For noise-free data, the errors for the inverse parameters and the

solution ﬁeld decrease exponentially as the number of training collocation points or the number of trainable

parameters in the neural network increases. These errors can reach a level close to the machine accuracy

when the simulation parameters become large. For noisy data, the current method remains quite accurate,

although the accuracy degrades compared with the case of noise-free data. We observe that, by scaling the

measurement-residual vector by a factor, one can markedly improve the accuracy of the current method for

noisy data, while only slightly degrading the accuracy for noise-free data. We have compared the current

method with the PINN method (see Appendix C). The current method exhibits an advantage in terms of

the accuracy and the computational cost (network training time).

The method and algorithms developed herein are implemented in Python based on the Tensorﬂow

(https://www.tensorﬂow.org/), Keras (https://keras.io/), and the scipy (https://scipy.org/) libraries. The

numerical simulations are performed on a MAC computer (3.2GHz Intel Core i5 CPU, 24GB memory) in

the authors’ institution.

The main contribution of this paper lies in the local extreme learning machine based technique together

with the three algorithms for solving inverse PDE problems. The exponential convergence behavior exhib-

ited by the current method for inverse problems is particularly interesting, and can be analogized to the

observations in [16] for forward PDEs. For inverse problems such fast convergence seems not available in the

existing techniques (e.g. PINN based methods).

The rest of this paper is structured as follows. In Section 2 we ﬁrst discuss the representation of functions

by local randomized neural networks and domain decomposition, and then present the NLLSQ, VarPro-F1

and VarPro-F2 algorithms for training the neural network to solve the inverse PDE. Section 3 uses a number

of inverse parametric PDEs to demonstrate the exponential convergence and the accuracy of our method, as

well as the eﬀects of the noise and the number of measurement points. Section 4 concludes the discussion with

some closing remarks. Appendix A summarizes the NLLSQ-perturb algorithm from [16] (with modiﬁcations),

which forms the basis for the three algorithms in the current paper for solving inverse PDEs. Appendix B

provides the matrices in the VarPro-F2 algorithm. Appendix C compares the current method with PINN

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domain

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sub-domain #1

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sub-domain #N

Figure 1: Cartoon illustrating domain decomposition and local random-weight neural networks.

for several inverse problems from Section 3. Appendix D lists the parameter values in the NLLSQ-perturb

algorithm for all the numerical simulations in Section 3.

2 Algorithms for Inverse PDEs with Randomized Neural Net-

works

2.1 Inverse Parametric PDEs and Local Randomized Neural Networks

We focus on the inverse problem described by the following parametric PDE, boundary conditions, and

measurement operations on some domain Ω ⊂Rd(d= 1,2,3):

α1L1(u) + α2L2(u) + · · · +αnLn(u) + F(u) = f(x),x∈Ω,(1a)

Bu(x) = g(x),x∈∂Ω,(1b)

Mu(ξ) = S(ξ),ξ∈Ωs⊂Ω.(1c)

In this system, Li(1 6i6n) and Fare diﬀerential or algebraic operators, which can be linear or nonlinear,

and fand gare prescribed source terms. u(x) is an unknown scalar ﬁeld, where xdenotes the coordinates.

αi(1 6i6n) are nunknown constants. The case with any αibeing an unknown ﬁeld distribution will be

dealt with later in a remark (Remark 2.7). We assume that the highest derivative term in (1a) is linear with

respect to u, while the nonlinear terms with respect to uinvolve only lower derivatives (if any). Bis a linear

diﬀerential or algebraic operator, and Budenotes the boundary condition(s) on the domain boundary ∂Ω.

Mis a linear algebraic or diﬀerential operator representing the measurement operations. Mu(ξ) denotes

the measurement of Muat the point ξ, and S(ξ) denotes the measurement data. Ωsdenotes the set of

measurement points. Given S(ξ), the goal here is to determine the parameters αi(1 6i6n) and the

solution ﬁeld u(x). Hereafter we will refer to the parameters α= (α1, . . . , αn)Tas the inverse parameters.

Suppose the inverse parameters are given. The boundary value problem consisting of the equations (1a)–(1b)

will be referred to as the associated forward PDE problem, with u(x) as the unknown. We assume that the

formulation is such that the forward PDE problem is well-posed.

Remark 2.1. We assume that the operators Li(16i6n) or Fmay contain time derivatives (e.g. ∂

∂t ,

∂2

∂t2, where tdenotes time), thus leading to an initial-boundary value problem on a spatial-temporal domain

Ω. In this case, we treat tin the same way as the spatial coordinate x, and use the last dimension in

x= (x1, x2, . . . , xd)to denote t(i.e. xd≡t). Accordingly, we assume that the equation (1b) should include

conditions on the appropriate initial boundaries from ∂Ω. The point here is that the system (1) may refer to

time-dependent problems, and we will not distinguish this case in subsequent discussions.

We devise numerical algorithms to compute a least squares solution to the system (1) based on local

randomized neural networks (or ELM). We decompose the domain Ω into sub-domains, and represent u(x)

on each sub-domain by a local ELM in a way analogous to in [16]. Let Ω = Ω1∪Ω2∪ · · · ∪ ΩN,where Ωi

(1 6i6N) denote Nnon-overlapping sub-domains (see Figure 1 for an illustration). Let

u(x) = 









u1(x),x∈Ω1,

u2(x),x∈Ω2,

. . .

uN(x),x∈ΩN,

(2)

where ui(x) (1 6i6N) denotes the solution ﬁeld restricted to the sub-domain Ωi. On the interior sub-

domain boundaries shared by adjacent sub-domains we impose Ckcontinuity conditions on u(x), where

k= (k1, . . . , kd) denotes a set of appropriate non-negative integers related to the order of the PDE (1a). If

the PDE order (highest derivative) is mialong the xi(1 6i6d) direction, we would in general impose

Cmi−1(i.e. ki=mi−1) continuity conditions in this direction on the shared sub-domain boundaries.

On Ωi(1 6i6N) we employ a local FNN, whose hidden-layer coeﬃcients are randomly assigned and

ﬁxed, to represent ui(x). More speciﬁcally, the local neural network is set as follows. The input layer consists

of dnodes, representing the input coordinate x= (x1, x2, . . . , xd)∈Ωi. The output layer consists of a single

node, representing ui(x). The network contains (L−1) (with integer L>2) hidden layers in between. Let

σ:R→Rdenote the activation function for all the hidden nodes. Hereafter we use the following vector (or

list) Mof (L+ 1) positive integers to represent the architecture of the local NN,

M= [m0, m1,...,mL−1, mL],(architectural vector) (3)

where m0=dand mL= 1 denote the number of nodes in the input/output layers respectively, and miis

the number of nodes in the i-th hidden layer (1 6i6L−1). We refer to Mas an architectural vector.

We make the following assumptions:

•The output layer should contain (i) no bias, and (ii) no activation function (or equivalently, the acti-

vation function be σ(x) = x).

•The weights/biases in all the hidden layers are pre-set to uniform random values on [−Rm, Rm], where

Rm>0 is a user-provided constant. The hidden-layer coeﬃcients are ﬁxed once they are set.

•The output-layer weights constitute the the trainable parameters of the local neural network.

We employ the same architecture, same activation function, and the same Rmfor the local neural networks

on diﬀerent sub-domains.

In light of these settings, the logic in the output layer of the local NNs leads to the following relation on

the sub-domain Ωi(1 6i6N),

ui(x) =

j=1

βij φij (x) = Φi(x)βi,(4)

where M=mL−1denotes the width of the last hidden layer of the local NN, φij (x) (1 6j6M) denote

the set of output ﬁelds of the last hidden layer on Ωi,βij (1 6j6M) denote the set of output-layer

coeﬃcients (trainable parameters) on Ωi, and Φi= (φi1, φi2, . . . , φiM ) and βi= (βi1, βi2, . . . , βiM )T. Note

that, once the random hidden-layer coeﬃcients are assigned, Φi(x) in (4) denotes a set of random (but ﬁxed

and known) nonlinear basis functions. Therefore, with local ELMs the output ﬁeld on each sub-domain is

represented by an expansion of a set of random basis functions as given by (4).

With domain decomposition and local ELMs, the system (1) is symbolically transformed into the following

form, which includes the continuity conditions across shared sub-domain boundaries:

α1L1(ui) + α2L2(ui) + · · · +αnLn(ui) + F(ui) = f(x),x∈Ωi,16i6N; (5a)

Bui(x) = g(x),x∈∂Ω∩Ωi,16i6N; (5b)

Mui(ξ) = S(ξ),ξ∈Ωs∩Ωi,16i6N; (5c)

Cui(x)− Cuj(x) = 0,x∈∂Ωi∩∂Ωj,for all adjacent sub-domains (Ωi,Ωj),16i, j 6N. (5d)

In this system ui(x) is given by (4), and the operator Cudenotes the set of Ckcontinuity conditions imposed

across the shared sub-domain boundaries on uor its derivatives. Deﬁne the residual of this system as,

R(α,β,x,ξ) = 





α1L1(ui) + α2L2(ui) + · · · +αnLn(ui) + F(ui)−f(x),x∈Ωi,16i6N

Bui(x)−g(x),x∈∂Ω∩Ωi,16i6N

Mui(ξ)−S(ξ),ξ∈Ωs∩Ωi,16i6N

Cui(x)− Cuj(x),x∈∂Ωi∩∂Ωj,for all adjacent (Ωi,Ωj),16i, j 6N





,(6)

where βis the vector of all trainable parameters, β= (βT

1,...,βT

N)T= (β11, β12, . . . , β1M, β21, . . . , βNM )T.

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AMethodforComputingInverseParametricPDEProblemswithRandom-WeightNeuralNetworksSuchuanDong*,YiranWangCenterforComputationalandAppliedMathematicsDepartmentofMathematicsPurdueUniversity,USA(October8,2022)AbstractWepresentamethodforcomputingtheinverseparametersandthesolutioneldtoinverseparametricpartia...

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A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks Suchuan Dong Yiran Wang

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