A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs Songming Liu1 Zhongkai Hao1 Chengyang Ying1 Hang Su12 Jun Zhu12 Ze Cheng3

2025-04-27 0 0 899.4KB 34 页 10玖币

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A Uniﬁed Hard-Constraint Framework for

Solving Geometrically Complex PDEs

Songming Liu1, Zhongkai Hao1, Chengyang Ying1, Hang Su1,2∗

, Jun Zhu1,2∗, Ze Cheng3

1Dept. of Comp. Sci. and Tech., Institute for AI, THBI Lab, BNRist Center,

Tsinghua-Bosch Joint ML Center, Tsinghua University

2Peng Cheng Laboratory; Pazhou Laboratory (Huangpu), Guangzhou, China

3Bosch Center for Artiﬁcial Intelligence

csuastt@gmail.com

Abstract

We present a uniﬁed hard-constraint framework for solving geometrically complex

PDEs with neural networks, where the most commonly used Dirichlet, Neumann,

and Robin boundary conditions (BCs) are considered. Speciﬁcally, we ﬁrst intro-

duce the “extra ﬁelds” from the mixed ﬁnite element method to reformulate the

PDEs so as to equivalently transform the three types of BCs into linear equations.

Based on the reformulation, we derive the general solutions of the BCs analytically,

which are employed to construct an ansatz that automatically satisﬁes the BCs.

With such a framework, we can train the neural networks without adding extra

loss terms and thus efﬁciently handle geometrically complex PDEs, alleviating

the unbalanced competition between the loss terms corresponding to the BCs and

PDEs. We theoretically demonstrate that the “extra ﬁelds” can stabilize the training

process. Experimental results on real-world geometrically complex PDEs showcase

the effectiveness of our method compared with state-of-the-art baselines.

1 Introduction

Many fundamental problems in science and engineering (e.g., [

]) are characterized by partial

differential equations (PDEs) with the solution constrained by boundary conditions (BCs) that are

derived from the physical system of the problem. Among all types of BCs, Dirichlet, Neumann, and

Robin are the most commonly used [

]. Figure 1 gives an illustrative example of these three types

of BCs. Furthermore, in practical problems, physical systems can be very geometrically complex

(where the geometry of the deﬁnition domain is irregular or has complex structures, e.g., a lithium-ion

battery [

], a heat sink [

], etc), leading to a large number of BCs. How to solve such PDEs has

become a challenging problem shared by both scientiﬁc and industrial communities.

The ﬁeld of solving PDEs with neural networks has a history of more than 20 years [

Such methods are intrinsically mesh-free and therefore can handle high-dimensional as well as

geometrically complex problems more efﬁciently compared with traditional mesh-based methods,

like the ﬁnite element method (FEM). Physical-informed neural network (PINN) [

] is one of the

most inﬂuential works, where the neural network is trained in the way of taking the residuals of

both PDEs and BCs as multiple terms of the loss function. Although there are many wonderful

improvements such as DPM [14], PINNs still face serious challenges as discussed in the paper [16].

Some theoretical works [

] point out that there exists an unbalanced competition between the

terms of PDEs and BCs, limiting the application of PINNs to geometrically complex problems. To

address this issue, some researchers [

] have tried to embed BCs into the ansatz. Some of

∗Corresponding author

36th Conference on Neural Information Processing Systems (NeurIPS 2022).

arXiv:2210.03526v6 [cs.LG] 4 Jun 2023

HEAT

ENVIRONMENT

Dirichlet BC Neumann BC Robin BC

Figure 1: An illustration on three types of BCs. We give an example from heat transfer, where

T=T(x)

is the temperature,

is the thermal conductivity, and

is the heat transfer coefﬁcient. (1)

The Dirichlet BC speciﬁes the value of the solution at the boundary. Here we assume a constant

temperature

THEAT

at the heat source (

x=x0

). (2) The Neumann BC speciﬁes the value of the

derivative at the boundary. We assume that the right wall (

x=x2

) is adiabatic and impose a zero

normal derivative. (3) The Robin BC is a combination of the ﬁrst two. And we use it to describe the

heat convection between the heat source and the environment (T∞) at the surface (x=x2).

them [

] follow the pipeline of the Theory of Connections [

], while others [

] have

considered solving the equivalent variational form of the PDEs. In this way, the neural networks can

automatically satisfy the BCs and no longer require adding corresponding loss terms. Nevertheless,

these methods are only applicable to speciﬁc BCs (e.g., Dirichlet BCs, homogeneous BCs, etc) or

geometrically simple PDEs. The key challenge is that the equation forms of the Neumann and Robin

BCs have no analytical solutions in general and are thus difﬁcult to be embedded into the ansatz.

In this paper, we propose a uniﬁed hard-constraint framework for all the three most commonly used

BCs (i.e., Dirichlet, Neumann, and Robin BCs). With this framework (see Figure 2 for an illustration),

we are able to construct an ansatz that automatically satisﬁes the three types of BCs. Therefore, we

can train the model without the losses of these BCs, which alleviates the unbalanced competition

and signiﬁcantly improves the performance of solving geometrically complex PDEs. Speciﬁcally,

we ﬁrst introduce the extra ﬁelds from the mixed ﬁnite element method [

]. This technique

substitutes the gradient of a physical quantity with new variables, allowing the BCs to be reformulated

as linear equations. Based on this reformulation, we derive a general continuous solution of the

BCs of simple form, overcoming the challenge that the original BCs cannot be solved analytically.

Using the general solutions obtained, we summarize a paradigm for constructing the hard-constraint

ansatz under time-dependent, multi-boundary, and high-dimensional cases. Besides, in Section 4, we

demonstrate that the technique of extra ﬁelds can improve the stability of the training process.

We empirically demonstrate the effectiveness of our method through three parts of experiments.

Firstly, we show the potency of our method in solving geometrically complex PDEs through two

numerical experiments from real-world physical systems of a battery pack and an airfoil. And

our framework achieves a supreme performance compared with advanced baselines, including the

learning rate annealing methods [

], domain decomposition-based methods [

], and existing

hard-constraint methods [

]. Second, we select a high-dimensional problem to demonstrate that

our framework can be well applied to high-dimensional cases. Finally, we study the impact of the

extra ﬁelds as well as some hyper-parameters and verify our theoretical results in Section 4.

To sum up, we make the following contributions:

•

We introduce the extra ﬁelds to reformulate the PDEs, and theoretically demonstrate that

our reformulation can effectively reduce the instability of the training process.

•

We propose a uniﬁed hard-constraint framework for Dirichlet, Neumann, and Robin BCs,

alleviating the unbalanced competition between losses in physics-informed learning.

•

Our method has superior performance over state-of-the-art baselines on solving geometri-

cally complex PDEs, as validated by numerical experiments in real-world physical problems.

Boundaries

Boundary Conditions

where is the unknown function

Extra fields

Reformulated Boundary Conditions

General Solutions

where is a neural network

Proposed Ansatz

Eq. (11)

Approximation Range of NNs

Ansatz which satisfies

Boundary Conditions

Unbalanced competition

Hypothesis space to explore

with only slightly relaxation

Deduce

Figure 2: A pipeline of the proposed method. In this paper, we consider PDEs with multiple

boundary conditions (BCs) of Dirichlet, Neumann, and Robin. We ﬁrst introduce the extra ﬁelds to

reformulate the BCs as linear equations whose general solutions are deduced. Then, we aggregate the

general solutions for each boundary to obtain our ansatz via Eq.

(11)

. Since the ansatz automatically

satisﬁes the BCs, we alleviate the unbalanced competition and reduce invalid hypothesis space.

2 Background

2.1 Physics-Informed Neural Networks (PINNs)

We consider the following Laplace’s equation as a motivating example:

∆u(x1, x2)=0, x1∈(0,1], x2∈[0,1],(1a)

u(x1, x2) = g(x2), x1= 0, x2∈[0,1],(1b)

where

g:R→R

is a known ﬁxed function, Eq.

(1a)

gives the form of the PDE, and Eq.

(1b)

is a

Dirichlet boundary condition (BC). A solution to the above problem is a solution to Eq.

(1a)

which

also satisﬁes Eq. (1b).

Physics-informed neural networks (PINNs) [

] employ a neural network

NN(x1, x2;θ)

to approx-

imate the solution, i.e.,

ˆu(x1, x2;θ) = NN(x1, x2;θ)≈u(x1, x2)

, where

denotes the trainable

parameters of the network. And we learn the parameters

by minimizing the following loss function:

L(θ) = LF(θ) + LB(θ)≜1

i=1 ∆ˆu(x(i)

f,1, x(i)

f,2;θ)

2+1

i=1 ˆu(0, x(i)

b,2;θ)−g(x(i)

b,2)

2,(2)

where

is the term restricting

ˆu

to satisfy the PDE (Eq.

(1a)

) while

is the one for the BC

(Eq.

(1b)

{x(i)

f= (x(i)

f,1, x(i)

f,2)}Nf

i=1

is a set of

collocation points sampled in

[0,1]2

, and

{x(i)

(0, x(i)

b,2)}Nb

i=1 is a set of Nbboundary points sampled in x1= 0 ∧x2∈[0,1].

PINNs have a wide range of applications, including heat [

], ﬂow [

], and atmosphere [

]. However,

PINNs are struggling with some issues on the performance [

]. Previous analysis [

] has

demonstrated that the convergence of

can be signiﬁcantly faster than that of

. This pathology

may lead to nonphysical solutions which do not satisfy the BCs or initial conditions (ICs). Moreover,

for geometrically complex PDEs where the number of BCs is large, this problem is exacerbated and

can seriously affect accuracy, as supported by our experimental results in Table 1.

2.2 Hard-Constraint Methods

One potential approach to overcome this pathology is to embed the BCs into the ansatz in a way

that any instance from the ansatz can automatically satisfy the BCs, as utilized by previous works

[

]. We note that the loss terms corresponding to the BCs are no longer needed,

and thus the above pathology is alleviated. These methods are called hard-constraint methods, and

they share a similar formula of the ansatz as:

ˆu(x;θ) = u∂Ω(x) + l∂Ω(x)NN(x;θ),(3)

where

is the coordinate,

Ω

is the domain of interest,

u∂Ω(x)

is the general solution at the boundary

∂Ω, and l∂Ω(x)is an extended distance function which satisﬁes:

l∂Ω(x) = 0 if x∈∂Ω,

>0 otherwise.(4)

In the case of Eq.

(1)

(where

x= (x1, x2)

Ω = [0,1]2

), the general solution is exactly

g(x2)

, and

we can use the following ansatz (which automatically satisﬁes the BC in Eq. (1b)):

ˆu(x1, x2;θ) = g(x2) + x1NN(x1, x2;θ).(5)

However, it is hard to directly extend this method to more general cases of Robin BCs (see Eq.

(7)

since we cannot obtain the general solution

u∂Ω(x)

analytically. Existing attempts are either mesh-

dependent [

], which are time-consuming for high-dimensional and geometrically complex

PDEs, or ad hoc methods for speciﬁc (geometrically simple) physical systems [

]. It is still lacking

a uniﬁed hard-constraint framework for both geometrically complex PDEs and the most commonly

used Dirichlet, Neumann, and Robin BCs.

3 Methodology

We ﬁrst introduce the problem setup of geometrically complex PDEs considered in this paper and

then reformulate the PDEs via the extra ﬁelds, followed by presenting our uniﬁed hard-constraint

framework for embedding Dirichlet, Neumann, and Robin BCs into the ansatz.

3.1 Problem Setup

We consider a physical system governed by the following PDEs deﬁned on a geometrically complex

domain: Ω⊂Rd

F[u(x)] = 0,x= (x1, . . . , xd)∈Ω,(6)

where

F= (F1,...,FN)

includes

PDE operators which map

to a function of

and

’s

derivatives. Here,

u(x)=(u1(x), . . . , un(x))

is the unknown solution, which represents physical

quantities of interest. For each uj, j = 1, . . . , n, we impose suitable boundary conditions (BCs) as:

aj,i(x)uj+bj,i(x)nj,i(x)· ∇uj=gj,i(x),x∈γj,i,∀i= 1, . . . , mj,(7)

where

{γj,i}mj

i=1

are subsets of the boundary

∂Ω

whose closures are disjoint,

aj,i, bj,i :γj,i →R

satisfy that

j,i(x) + b2

j,i(x)̸= 0,∀x∈γj,i

nj,i :γj,i →Rd

is the (outward facing) unit normal

γj,i

at each location, and

gj,i :γj,i →R

. It is noted that Eq.

(7)

represents a Dirichlet BC if

aj,i ≡1, bj,i ≡0

, a Neumann BC if

aj,i ≡0, bj,i ≡1

, and a Robin BC otherwise. In the following,

we drop some of the subscripts in Eq. (7) for clarity 2.

For such geometrically complex PDEs, if we directly resort to PINNs (see Section 2.1), there would

be a difﬁcult multi-task learning with at least

(Pn

j=1 mj+N)

terms in the loss function. As discussed

in the previous analyses [

], it will severely affect the convergence of the training due to the

unbalanced competition between those loss terms. Hence, in this paper, we will discuss how to

embed the BCs into the ansatz, where every instance automatically satisﬁes the BCs. However, it is

infeasible to directly follow the pipeline of hard-constraint methods (see Eq.

(3)

) since Eq.

(7)

does

not have a general solution of analytical form. Therefore, a new approach is needed to address this

intractable problem.

3.2 Reformulating PDEs via Extra Fields

In this subsection, we present the general solutions of the BCs, which will be used to construct

the hard-constraint ansatz subsequently. We ﬁrst introduce the extra ﬁelds from the mixed ﬁnite

element method [

] to equivalently reformulate the PDEs. Let

pj(x)=(pj1(x), . . . , pjd(x)) =

∇uj, j = 1, . . . , n. We substitute them into Eq. (6) and Eq. (7) to obtain the equivalent PDEs:

F[u(x),p1(x),...,pn(x)] = 0,x∈Ω,(8a)

pj(x) = ∇uj,x∈Ω∪∂Ω,∀j= 1, . . . , n, (8b)

2We simplify γj,i , aj,i(x), bj,i(x),nj,i(x), gj,i(x)to γi, ai(x), bi(x),n(x), gi(x).

where

(u(x),p1(x),...,pn(x))

is the solution to the new PDEs,

F= ( ˜

F1,..., ˜

FN)

are the PDE

operators after the reformulation. And for j= 1, . . . , n, we have the corresponding BCs:

ai(x)uj+bi(x)n(x)·pj(x)=gi(x),x∈γi,∀i= 1, . . . , mj.(9)

Now, we can see that Eq.

(7)

has been transformed into linear equations with respect to

(uj,pj)

which are much easier for us to derive general solutions. Hereinafter, we denote

(uj,pj)

, and

the general solution to the BC at γiby ˜

pγi

j= (uγi

j,pγi

j). Next, we will discuss how to obtain ˜

pγi

To obtain the general solution of Eq.

(9)

, the ﬁrst step is to ﬁnd a basis

B(x)

of the null space (whose

dimension is

), which should include

vectors. However, we must emphasize that

B(x)

should be

carefully chosen. Since Eq.

(9)

is parameterized by

, for any

x∈γi

B(x)

should always be a basis

of the null space, that is, its columns cannot degenerate into linearly dependent vectors (otherwise it

will not be able to represent all possible solutions). An example of an inadmissible

B(x)

is given in

Appendix A.1.

Generally, for any dimension

d∈N+

, we believe it is non-trivial to ﬁnd a simple expression for the

basis. Instead, we prefer to ﬁnd

(d+ 1)

vectors in the null space,

of which are linearly independent

(that way,

B(x)∈R(d+1)×(d+1)

). We now directly present our construction of the general solution

while leaving a detailed derivation in Appendix A.3.

pγi

j(x;θγi

j) = B(x)NNγi

j(x;θγi

j) + ˜

n(x)˜gi(x),(10)

where

n= (ai, bin)pa2

i+b2

˜gi=gipa2

i+b2

NNγi

j:Rd→Rd+1

is a neural network with

trainable parameters θγi

j, and B(x) = Id+1 −˜

n(x)˜

n(x)⊤is the basis we have found (precisely, it

is a set of vectors that always contain a basis of the null space for any

x∈γi

). Incidentally, in the

case of

d= 1

d= 2

, we can ﬁnd a simpler expression for

(see Appendix A.2). We note that

there is no restriction for the architecture of neural networks used and MLPs are chosen as default.

3.3 A Uniﬁed Hard-Constraint Framework

With the parameterization of a neural network, Eq.

(10)

can represent any function deﬁned on

γi

as long as the function satisﬁes the BC (see Eq.

(9)

). Since our problem domain contains multiple

boundaries, we need to combine the general solutions corresponding to each boundary

γi

to achieve

an overall approximation. Hence, we construct our ansatz (for each

1≤j≤n

separately) as follows:

(ˆuj,ˆ

pj) = l∂Ω(x)NNmain

j(x;θmain

j) +

i=1

exp −αilγi(x)˜

pγi

j(x;θγi

j),∀j= 1, . . . , n, (11)

where

NNmain

j:Rd→Rd+1

is the main neural network with trainable parameters

θmain

l∂Ω, lγi, i = 1, . . . , mj

are continuous extended distance functions (similar to Eq.

(4)

), and

αi(i= 1, . . . , mj)are determined by:

αi=βs

minx∈∂Ω\γilγi(x),(12)

where

βs∈R

is a hyper-parameter of the “hardness” in the spatial domain. In Eq.

(11)

, we utilize

extended distance functions to “divide” the problem domain into several parts, where

{˜

pγi

j}mj

i=1

is responsible for the approximation on the boundaries while

NNmain

are responsible for internal.

Furthermore, Eq.

(12)

ensures that the importance of

pγi

decays to

e−βs

on the nearest boundary

from

γi

, so that

pγi

does not appear on other boundaries. We provide a theoretical guarantee for the

correctness and approximation ability of Eq.

(11)

in Appendix A.4. Besides, if

are

only deﬁned at

γi

, we can extend their deﬁnition to

Ω∪∂Ω

using interpolation techniques or neural

networks (see Appendix A.5). An extension of our framework to the temporal domain is discussed in

Appendix A.6.

Finally, we can train our model with the following loss function:

L=1

k=1

j=1 ˜

Fj[ˆ

u(x(k)),ˆ

p1(x(k)),..., ˆ

pn(x(k))]

k=1

j=1 

ˆ

pj(x(k))− ∇ˆuj(x(k))



(13)

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AUnifiedHard-ConstraintFrameworkforSolvingGeometricallyComplexPDEsSongmingLiu1,ZhongkaiHao1,ChengyangYing1,HangSu1,2∗,JunZhu1,2∗,ZeCheng31Dept.ofComp.Sci.andTech.,InstituteforAI,THBILab,BNRistCenter,Tsinghua-BoschJointMLCenter,TsinghuaUniversity2PengChengLaboratory;PazhouLaboratory(Huangpu),Guangzho...

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A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs Songming Liu1 Zhongkai Hao1 Chengyang Ying1 Hang Su12 Jun Zhu12 Ze Cheng3

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