A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs
Anonymous Author(s)
AffiliationAddress email
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 introduce the “extra fields” from the mixed finite element method to reformulatethe PDEs so as to equivalently transform the three types of BCs into linear forms.
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 extraloss terms and thus efficiently handle geometrically PDEs with complicated BCs,alleviating the unbalanced competition between the loss terms corresponding to the
BCs and PDEs.
We theoretically demonstrate that the “extra fields” can stabilize thetraining process.
Experimental results on real-world geometrically complex PDEsshowcase the effeteness of our method compared with state-of-the-art baselines.
Introduction
Many fundamental problems in science and engineering (e.g., [2, 17, 26]) are characterized by partialdifferential equations (PDEs) with the solution constrained by boundary conditions (BCs) that arederived from the physical system of the problem.
Among all types of BCs, Dirichlet, Neumann and
Robin are the most commonly used[31].
Figure 1 gives an illustrative example on these three typesof BCs.
Furthermore, in practical problems, physical systems can be very geometrically complex(e.g., a lithium-ion battery [10], a heat sink [38], etc), leading to a large number of complex BCs in PDEs.
How to solve such PDEs has become a challenging problem shared by both the scientific andindustrial communities.
Recently, a new line of work [23, 1, 32, 7, 34] introduces neural networks in solving PDEs.
Suchmethods 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 thefinite element method (FEM).
Physical-informed neural network (PINN) [23] is one of the mostinfluential works, where the neural network is trained in the way of taking the residuals of both the PDEs and the BCs as multiple terms of the loss function.
This multi-task learning type scheme iswell applied in experiments but still at a very early stage [13].
Some theoretical work [35, 36] pointsout that there exists an unbalanced competition between the terms of PDEs and BCs, limiting theapplication of PINNs to geometrically complex problems.
To address this issue, some researchers[3, 33, 16] have tried to embed BCs into the ansatz.
Some of them [14, 25] follow the pipeline ofthe Theory of Connections [20], while others [39, 11, 15] have considered solving the equivalentvariational form of the PDEs.
In this way, the neural networks can automatically satisfy the BCs andno longer require adding corresponding loss terms.
Nevertheless, these methods are only applicableto specific BCs (e.g., Dirichlet BCs, homogeneous BCs, etc) or geometrically simple PDEs. The key
[Figure]
challenge is that the equation forms of the Neumann and Robin BCs have no analytical solutions ingeneral 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, we are able to construct anansatz that automatically satisfies the three types of BCs.
Therefore, we can train the model withoutthe losses of these BCs, which alleviates the unbalanced competition and significantly improves theperformance of solving geometrically complex PDEs.
Specifically, we first introduce the extra fieldsfrom the mixed finite element method [18, 4].
This technique substitutes the gradient of a physicalquantity with new variables, allowing the BCs to be reformulated as linear equations.
Based on thisreformulation, we derive a general continuous solution of the BCs with simple form, overcoming thechallenge 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, multiboundary, and high-dimensional cases.
Besides, in Section 4, we demonstrate that the technique ofextra fields can improve the stability of the training process.
We empirically demonstrate the effectiveness of our method through three parts of experiments.
First,we show the potency of our method in solving geometrically complex PDEs through two numericalexperiments from real-world physical systems of a battery pack and an airfoil.
And our frameworkachieves a supreme performance compared with advanced baselines, including the learning rateannealing method [35], the methods based on domain-decomposition [9, 21], and existing hardconstraint methods [28, 29].
Second, we select a high-dimensional problem to demonstrate that ourframework can be well applied to high-dimension cases.
Finally, we study the impact of the extrafields 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 thatour reformulation can effectively reduce the instability of the training process.•
We propose a unified hard-constraint framework for Dirichlet, Neumann, and Robin BCs,alliviating the unbalanced competition between losses in the physics-informed learning.•
Our method has superior performance over state-of-the-art baselines on solving geometrically complex PDEs, as validated by numerical experiments in real-world physical problems.
Background2.1 Physics-Informed Neural Networks (PINNs)
We consider the following Laplace’s equation as a motivating example,
[Equation]
where Eq. (1a) gives the form of the PDE, and Eq.
(1b) is a Dirichlet boundary condition (BC).
Asolution to the above problem is a solution to Eq.
(1a) which also satisfies Eq.
(1b).
Physics-informed neural networks (PINNs) [23] employ a neural network NN( x 1 , x 2 ; θ ) to approximate the solution, i.e., ˆ u ( x 1 , x 2 ; θ )
=
NN( x 1 , x 2 ; θ ) ≈ u ( x 1 , x 2 ) , where θ denotes the trainableparameters of the network.
And we learn the parameters θ by minimizing the following loss function
[Equation]
where L F is the term restricting ˆ u to satisfy the PDE (Eq. (1a)) while L B is the BC (Eq. (1b))
, { x ( i ) f =( x
( i ) f, 1 , x ( i ) f, 2 ) } N f i
=1 is a set of N f collocation points sampled from [0 , 1] 2 , and { x ( i ) b = (0 , x ( i ) b, 2 ) } N b i =1 is aset of N b collocation points sampled from x 1 = 0 .
PINNs have a wide range of applications, including heat [5], flow [19], and atmosphere [40].
However,
PINNs are struggling with some issues on the performance [13].
Previous analysis [35, 36] hasdemonstrated that the convergence of L F can be significantly faster than that of L B .
As demonstratedin later experiments (see Section 5), this pathology may lead to nonphysical solutions which does notsatisfy the BCs or initial conditions (ICs).
Moreover, for geometrically complex PDEs where thenumber of BCs is large, this problem is exacerbated and can seriously affect accuracy, as supportedby 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 waythat any instance from the ansatz can automatically satisfy the BCs, as utilized by previous work[3, 22, 33, 37, 28, 16, 29].
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 , andthey share a similar formula of the ansatz as
[Equation]
where x 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
[Equation]
In the case of Eq.
(1) (where x = ( x 1 , x 2 ) , Ω = [0 , 1] 2 ), the general solution is exactly g ( x 2 ) , andwe can use the following ansatz (which automatically satisfies the BC in Eq [Equation]
[Equation]
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 meshdependent [8, 41], which are time-consuming for high-dimensional and geometrically complex PDEs,or ad hoc methods for specific (geometrically simple) physical systems [24].
It is still lacking aunified hard-constraint framework for both geometrically complex PDEs and the most commonlyused Dirichlet, Neumann, and Robin BCs.
Methodology
[Figure] We first introduce the problem setup of geometrically complex PDEs considered in this paper andthen reformulate the PDEs via the extra fields , followed by presenting our unified hard-constraintframework for embedding Dirichlet, Neumann and Robin BCs into the ansatz.
3.1 Problem Setup
[Figure]
We consider a physical system governed by the following PDEs defined on a geometrically complexdomain Ω ⊂ R d
[Equation]
where F = ( F 1 , . . . , F N ) includes N PDE operators, and u ( x )
= ( u 1 ( x ) , . . .
, u n ( x )) is thesolution of the PDEs, which represents the physical quantities of interest.
For each u j , j = 1 , . . . , n ,we pose suitable boundary conditions (BCs) as
[Equation]
where γ 1 , . . . , γ m j are subsets of the boundary ∂ Ω whose closures are disjoint, a i ( x ) and
b i ( x ) aretwo functions satisfying that a 2 i ( x ) + b 2 i ( x ) ̸ = 0 holds for ∀ x ∈ γ i , and n ( x ) =
( n 1 ( x ) , . . . , n d ( x ))is the (outward facing) unit normal of corresponding γ
i at x .
It is noted that Eq.
(7) represents a
Dirichlet BC if a i ≡ 1 , b i ≡ 0 , a Neumann BC if a i ≡ 0 , b i ≡ 1 , and a Robin BC otherwise.
For such geometrically complex PDEs, if we directly resort to PINNs (see Section 2.1), there wouldbe a difficult multi-task learning with at least ( (cid:80) nj =1 m j + N ) terms in the loss function.
As discussedin the previous analysis [35, 36], it will severely affect the convergence of the training due to theunbalanced competition between those loss terms.
Hence, in this paper, we will discuss how toembed the BCs into the ansatz, where every instance automatically satisfies the BCs.
However, it isinfeasible to directly follow the pipeline of hard-constraint methods (see Eq.
(3)) since Eq.
(7) doesnot have an general solution of analytical form.
Therefore, a new approach is needed to address thisintractable problem.
3.2 Reformulating PDEs via Extra Fields
In this subsection, we present the general solutions of the BCs, which will be used to constructthe hard-constraint ansatz subsequently.
We first introduce the extra fields from the mixed finiteelement method [18, 4] to equivalently reformulate the PDEs.
Let p j
( x ) = ( p j 1 ( x ) , . . .
, p jd ( x ))
=∇ u j , j = 1 , . .
. , n .
And we substitute them into Eq.
(6) and Eq.
(7) to obtain the equivalent PDEs,˜ F [ u ( x ) , p 1 ( x ) , . . . , p n ( x )] = 0 , x [Equation] (8a) p
[Equation]
=
[Equation]
∀
[Equation]
= [Equation] [Equation] n, (8b) where ( u ( x ) , p 1 ( x ) , . . .
, p n ( x )) is the solution of the new PDEs, ˜ F =
( ˜ F 1 , . . .
, ˜ F N ) includes N PDE operators after the reformulation.
And for j = 1 , . . . , n , we have the corresponding BCs
[Equation]
Now, we can see that Eq.
(7) has been transformed into linear equations with respect to ( u j , p j ) . Based on the linear form, it is much easier for us to derive general solutions for both u j and p j ,instead of handling u j alone.
Hereinafter, we denote ( u j , p j ) by ˜ p
j , and their general solutions at γ i(i.e., ( u γ i j , p γ i j ) ) by ˜ p γ i j .
Next we will discuss how to find ˜ p γ i j .
To obtain the general solution of Eq.
(9), the first step is to find a basis B ( x ) of the null space (whosedimension is d ).
However, we must emphasize that B ( x ) should be carefully chosen.
Since Eq.
(9)is parameterized by x , for any x ∈ γ i , , B ( x ) should always be a basis of the null space, that is, itscolumns cannot degenerate into linearly dependent vectors (otherwise it will not be able to representall possible solutions).
An example of an inadmissible B ( x ) is given in Appendix A.1.
Generally, for any dimension d ∈ R , it is non-trival to find a simple expression.
Although we mightbe able to find an admissible one in a more complicated way (e.g, Gram–Schmidt orthogonalization),it would get expensive to evaluate.
Instead, we prefer to find ( d + 1) vectors of simple form in the null space, d of which are linearly independent (that way, B ( x ) ∈ R ( d +1) ×
( d +1) ).
Based on thisinsight, we determine the general solution as follows
[Equation]
where ˜ n = ( a i , b i n ) (cid:14)(cid:112) a 2 i + b 2 i , ˜ g
i =
g i (cid:14)(cid:112) a 2 i + b 2 i , NN i : R d → R d +1 is a neural network (herewe omit its trainable parameters), and B ( x ) =
I d +1 − ˜ n ( x )˜ n ( x ) ⊤ .
Incidentally, in the case ofd = 2 or d = 3 , we can find a simpler expression for B . See Appendix A.2 for more discussion.
3.3 A Unified Hard-constraint Framework
We now present our hard-constraint framework in detail. Based on the above reformulation andgeneral solutions, our ansatz (ˆ u j , ˆ p j ) is given by
[Equation]
where NN : R d → R d +1 is a neural network, l ∂ Ω , l γ
i , i = 1 , . . .
, m j are extended distance functions(see Eq.
(4)), and α i ( i = 1 , . . .
, m j ) are determined by
[Equation]
(12) where β s ∈ R is a hyper-parameter that determines the “hardness” in the spatial domain.
The aboveformula ensures that the weight of ˜ p γ i j ( x ) decays to e − β s at the nearest neighbor of γ i , so that ˜ p γ i j ( x )does not occur at the other boundaries.
Besides, if a i , b i , n or g i are only defined at γ i , we can extendtheir definition to Ω ∪ ∂ Ω using interpolation or approximation via neural networks. We put morediscussion in Appendix A.3.
What is more, although here we do not cover time-dependent PDEs, it iseasy to extend this framework to the spatial-temporal domain, where a hyper-parameter of “hardness”in the temporal domain β t is similarly defined. Refer to Appendix A.4 for more details.
Finally, we can train our model with the following loss function
[Equation]
where θ is the set of trainable parameters in all the neural networks,
ˆ u = (ˆ u 1 , . . .
, ˆ u n ) , { ˆ u j , ˆ p j } nj =1is defined in Eq.(11), and { x ( k ) } N f k
=1 is a set of collocation points sampled in Ω .
Eq.
(13) measuresthe discrepancy of both the PDEs (i.e., ˜ F 1 , . .
. , ˜ F N ) and the equilibrium equations introduced by theextra fields (i.e., Eq. (8b)) at N f collocation points.
According to Eq.
(13), we have now successfully embedded BCs into the ansatz, and no longer needto take the residuals of BCs as extra terms in the loss function.
That is, our ansatz strictly conformsto BCs throughout the training process, greatly reducing the possibility of generating nonphysicalsolutions.
Nevertheless, this comes at the cost of introducing ( nd ) additional equilibrium equations(see Eq.
(8b)).
But in many physical systems, especially those with complex geometries, the numberof BCs ( cnt(BCs) ) is far larger than ( nd ) (e.g., n = 3 , d = 2 , cnt(BCs)
=
1260 for a classicalphysical system, a heat exchanger [6]).
So we may actually reduce the number of loss terms byan order of magnitude or two ( ∆cnt(losses) = nd − cnt(BCs)
≪ 0 ), alleviating the unbalancedcompetition between loss terms.
Theoretical Analysis
In Section 3.2, we introduce the extra fields and reformulate the PDEs (from Eq.
(6) and Eq.
(7) to
Eq.
(8) and Eq.
To further analyze the impact of this reformulation, we consider the following abstraction of 1D PDEs (Eq. (14a)) and the one after the reformulation (Eq. (14b))
[Equation]
where L is an operator, f is a source function, and p i +
= d p i / d x, p 1 =
d u/ d x are the introducedextra fields.
From the above equations, we can find that the orders of derivatives in the PDEs arereduced.
Intuitively, as the PDEs are included in the loss function (see Eq.
(2)), lower derivativesresult in less accumulation of back-propagation, and thus stabilize the training process.
To formally explain this mechanism, without loss of generality, we consider a simple case (withproper Dirichlet BCs) where L [ · · · ] = ∆ u , f ( x ) = − a 2 sin ax .
And our ansatz is a single-layerneural network of length K , i.e., ˆ u =
c ⊤ g ( w x + b ) , ˆ p =
c ⊤ p g ( w x + b ) , where c , w , b , c p ∈ R K , gis an element-wise activation function (for simplicity, we take g as tanh ).
More details on the PDEsare given in Appendix A.5.
Next, we focus on the gradients of the loss terms of the PDEs F , sincethat of the BC stays the same during the reformulation.
We state the following theorem.
Theorem 4.1 (Bounds for the Gradients of the PDE Loss Terms) .
Let θ =
( c , w , b ) , ˜ θ =( c , w , b , c p ) , and L F as well as ˜ L F be the loss terms corresponding to the original and transformed PDEs, respectively.
We have the following bounds [Figure]
[Equation]
where ◦ is the element-wise multiplication, 1 is an all-ones vector, max( · ) is the element-wisemaximum of vectors, and operations on vectors ( |·| , ( · ) 2 , etc) are element-wise operations.
The proof is deferred to Appendix A.5.
We can find that ( ∇ θ L F ) ⊤ has a high-order relationship withthe parameters of the neural network θ .
This often makes training unstable, as the gradients ( ∇ θ L F ) ⊤may rapidly explode or vanish when θ are large or small.
After the reformulation, ( ∇ ˜ θ ˜ L F ) ⊤ isassociated with the parameters in a lower order, and some terms are controlled by 1 , alleviating thisissue of the instability.
We note that the improvement in stability comes from lower derivatives in the PDEs, and this mechanism does not depend on the specific form of the PDEs.
In Section 5.5.1, weempirically show that this mechanism is general, even for nonlinear PDEs.
Experiments
In this section, we will numerically showcase the effectiveness of the proposed framework ongeometrically complex PDEs, which is also the main focus of this paper.
To this end, we test ourframework on two real-world problems (see Section 5.2 and Section 5.3) from a 2D battery pack andan airfoil, respectively.
Besides, to validate that our framework is applicable to high-dimensionalcases, an addition high-dimension problem is considered in Section 5.4, followed by an ablationstudy in Section 5.5.
We refer to Appendix A.6 for experimental details.
[Figure]
5.1 Experiment Setup
Evaluation In our experiments, we utilize mean absolute error (MAE) and mean absolute percentage error (MAPE) to measure the discrepancy between the prediction and the ground truth.
In
Section 5.3, we replace MAPE with weighted mean absolute percentage error (WMAPE) to avoid the“division by zero”, since a considerable part of the ground truth values are very close to zero,
[Equation]
where n is the number of testing samples, ˆ y i is the prediction and y i is the ground truth.
Baselines We consider the following baselines in subsequent experiments.
• PINN : original implementation of the PINN [23].
• PINN-LA & PINN-LA-2 : PINN with learning rate annealing algorithm to address the
unbalanced competition [35] and a variant we modified for numerical stability.
• xPINN& FBPINN : two representative works for solving geometrically complex PDEs via
non-overlapping [9] and overlapping [21] domain decomposition.
• PFNN & PFNN-2 : a representative hard-constraint method based on the variational formu-
lation of PDEs [28] and its latest variant incorporating domain decomposition [29].
5.2 Simulation of a 2D battery pack (Heat Equation)
To emphasize the capability of the proposed hard-constraint framework (HC) to handle geometricallycomplex cases, we first consider modeling thermal dynamics of a 2D battery pack [30], which isgoverned by the heat equation (see Appendix A.6.1), where x ∈ Ω , t ∈ [0 , 1] are the spatial andtemporal coordinates, respectively, T ( x , t ) is the temperature over time.
The geometry of the problemΩ is shown in Figure 2(a), where γ cell ,i is the cell and γ pipe ,i is the cooling pipe.
The baselines are trained with N f = 8192 collocation points, N b = 512 boundary points and
N i =
512 initial points (an additional N s = 512 points on the interfaces between subdomains arerequired for the xPINN), while the HC is trained with N f = 8192 collocation points.
All the modelsare trained for 5000 Adam [12] iterations (with a learning rate scheduler) followed by a L-BFGSoptimization until convergence and tested with N t = 146 , 487 points evaluated by the FEM.
Thetesting results are given in Table 1 (where “average” means averaging over t ∈ [0 , 1] ).
We find thatthe accuracy of the HC is significantly higher than baselines almost all the time, especially at t = 0 .
This is because the ansatz of the HC always satisfies the BCs and ICs during the training process,prevents the approximations from violating physical constraints at the boundaries.
5.3 Simulation of an Airfoil (Navier-Stokes Equations)
In the next experiment, we consider the challenging benchmark PDEs in computational fluid dynamics,the 2D stationary incompressible Navier-Stokes equations, in the context of simulating the airflowaround a real-world airfoil ( w1015.dat ) from the UIUC airfoil coordinates database (an open airfoil
[Table] [Table]
database)
Specifically, the airfoil is represented by 240 anchor points and the governingequation can be found in Appendix A.6.2, where u ( x ) =
( u 1 ( x ) , u 2 ( x )) , p ( x ) are the velocity, andthe pressure of the fluid, respectively, and the geometry of the problem is shown in Figure 2(b).
The baselines (PFNN and PFNN-2 are not applicable to Navier-Stokes Equations) are trained with
N f = 10 , 000 collocation points, N b = 2048 boundary points ( N s = 2048 points for the xPINN),while the HC is trained with N f = 10 , 000 collocation points.
All the models are trained for
Adam iterations followed by a L-BFGS optimization until convergence and tested with N t = 13 , 651points evaluated by the FEM.
According to the results in Table 2, the HC has more obvious advantages,compared with the previous experiment.
This is because complex nonlinear PDEs are more sensitiveto the BCs, and the failure at the BCs can cause the approximations to deviate significantly from thetrue solutions.
In addition, we find that the WMAPE on u 2 is relatively large for all methods, whichis because the true values of u 2 are close to zeros, amplifying the effect of the noise.
5.4 High-dimensional Heat Equation
To demonstrate that our framework can generalize to high-dimensional cases, we consider a problemof time-dependent high-dimensional PDEs in Ω × [0 , 1] (see Appendix A.6.3), where Ω is a unit ballin R 10 , and u is the quantity of interest.
Here, we do not consider the baselines of the xPINN, FBPINN, and PFNN-2, because the methodof domain decomposition is not suitable for geometrically simple but high-dimensional cases.
Andthe other baselines are trained with N f = 1000 collocation points, N b = 100 boundary points and N i = 100 initial points, while the HC is trained with N f = 1000 collocation points.
All the modelsare trained for 5000 Adam iterations followed by a L-BFGS optimization until convergence and testedwith N t = 10 , 000 points evaluated by the analytical solution to the PDEs.
We refer to Table 3 for thetesting results.
It can be seen that the HC achieves impressive performance in high-dimensional cases.
This illustrates the effectiveness of Eq.
(10) and the fact that our model can be directly applied tohigh-dimensional PDEs, not just geometrically complex PDEs.
5.5 Ablation Study
In this subsection, we are interested in the impact of the “extra fields” and the hyper-parameters of“hardness” (i.e., β s and β t , see Section 3.3).
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We test the PINNs (with and without the “extra fields”) on the Poission equation and the nonlinear
Schrödinger equation (see Appendix A.6.4).
We vary the network architecture and report the ratio ofthe the moving variance (MovVar) of |∇ θ L F | to that of |∇ ˜ θ
˜ L F | at each iteration (see Figure 3).
Inboth linear and nonlinear PDEs, we find that models with the “extra fields” ( |∇ ˜ θ
˜ L F | ) achieve smallergradient oscillations during training, which is consistent with our theoretical results in Section 4.
This is because the “extra fields” reduces the orders of the derivatives, which in turn reduces theaccumulation of backpropagation, avoiding vanishing or exploding gradients.
5.5.2 Hyper-parameters of Hardness
We repeated the experiment in Section 5.2 with different combinations of the values of β s and β t ,and all other settings stay the same.
Table 4 gives the testing results (where the MAE and MAPEare the average results over t ∈ [0 , 1] ).
In general, the accuracy increases as the values of β s and β tbecome larger.
The reason is that “harder” β s and β t cause the BCs to be better satisfied, which inturn makes the approximations more compliant with physical constraints.
However, too large β s andβ t (they are both on the exponential, see Eq.(11)) can lead to performance degradation, which maybe related to numerical stability.
[Table]
Conclusion
In this paper, we develop a unified hard-constraint framework for solving geometrically complex
PDEs.
With the help of “extra fields”, we reformulate the PDEs and find the general solutions of the
Dirichlet, Neumann, and Robin BCs.
Based on this derivation, we propose a general formula of thehard-constraint ansatz which is applicable to time-dependent, multi-boundary, and high-dimensionalcases.
Besides, our theoretical analysis demonstrates that the reformulation is helpful for trainingstability.
Extensive experiments show that our method can achieve state-of-the-art performancein real-world geometrically complex as well as high-dimension PDEs, and our theoretical resultsare universal to general PDEs.
One of the limitations is that here we only consider the three mostcommonly used BCs and future works may include extending the framework to more general BCs.
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