Is L 2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?
Anonymous Author(s)
AffiliationAddress email
Abstract
The Physics-Informed Neural Network (PINN) approach is a new and promisingway to solve partial differential equations using deep learning.
The L 2 PhysicsInformed Loss is the de-facto standard in training Physics-Informed Neural Networks.
In this paper, we challenge this common practice by investigating therelationship between the loss function and the approximation quality of the learnedsolution.
In particular, we leverage the concept of stability in the literature of partialdifferential equation to study the asymptotic behavior of the learned solution asthe loss approaches zero.
With this concept, we study an important class of highdimensional non-linear PDEs in optimal control, the Hamilton-Jacobi-Bellman(HJB) Equation, and prove that for general L p Physics-Informed Loss, a wide classof HJB equation is stable only if p is sufficiently large.
Therefore, the commonlyused L 2 loss is not suitable for training PINN on those equations, while L ∞ loss isa better choice.
Based on the theoretical insight, we develop a novel PINN trainingalgorithm to minimize the L ∞ loss for HJB equations which is in a similar spiritto adversarial training.
The effectiveness of the proposed algorithm is empiricallydemonstrated through experiments.
Introduction
Recently, with the explosive growth of available data and computational resources, there have beengrowing interests in developing machine learning approaches to solve partial differential equations(PDEs) [14, 13, 33, 28].
One seminal work in this direction is the Physics-Informed Neural Network(PINN) approach [28] which parameterizes the PDE’s solution as a neural network.
By definingdifferentiable loss functionals that measure how well the model fits the PDE and boundary conditions,the network parameters can be efficiently optimized using gradient-based approaches.
L 2 distance isone of the most popularly used measures, which calculates the L 2 norm of the PDE and boundaryresidual on the domain and boundary, respectively.
Previous works demonstrated that PINN couldsolve a wide range of PDE problems using the L 2 Physics-Informed Loss, such as Poisson equation,
Burgers’ equation, and Navier-Stokes equation [28, 6].
Although previous works empirically demonstrated promising results using L 2 Physics-Informed
Loss, we argue the plausibility of using this loss for (high-dimensional) non-linear PDE problems.
We know the trivial fact that the learned solution will equal the exact solution when its L 2 loss equalszero.
However, the quality of a learned solution with a small but non-zero loss, which is a morerealistic scenario in practice, remains unknown to have any approximation guarantees.
In this work,we aim at answering a fundamental question:
Can we guarantee that a learned solution with a small Physics-Informed Loss always corresponds toa good approximator of the exact solution?
To thoroughly investigate the problem, we advocate analyzing the stability of PDE [8] in the PINNframework.
Stability characterizes the asymptotic behavior of the distance between the learnedsolution and the exact solution when the Physics-Informed Loss approaches zero.
If the PDE isnot stable with respect to certain loss functions, we may not obtain good approximate solutionsby minimizing the loss.
To show the strength of the theory, we perform a comprehensive studyon the stability of an important class of high-dimensional non-linear PDEs in optimal control, the
Hamilton-Jacobi-Bellman (HJB) equation, which establishes a necessary and sufficient condition fora control’s optimality with regard to the cost function.
Interestingly, we prove that for general L p
Physics-Informed Loss, the HJB equation is stable only if p is sufficiently large.
This finding suggeststhat the most widely used L 2 loss may not be suitable for training PINN on HJB equations as thelearned solution can be arbitrarily distant from the exact solution.
Empirical observation verifies thetheoretical results.
We further show our theory can serve as a principled way to design loss functions for training PINN.
For the high-dimensional HJB equation we target in the paper, the theoretical result suggests that L ∞loss may be a better choice to learn approximate solutions.
Motivated by this insight, we propose anew algorithm for training PINN, which adopts a min-max optimization procedure to minimize the
L ∞ loss.
Our approach resembles the well-known adversarial training framework.
In each iteration,we first fix the network parameters and learn adversarial data points to approximate L ∞ loss, andthen optimize the network parameters to minimize the loss.
When the training finishes, the learnednetwork will converge to a solution with small L ∞ losses and is close to the exact solution.
Weconduct experiments to demonstrate the effectiveness of the proposed algorithm.
All empirical resultsshow that our method can indeed learn accurate solutions for HJB equations and is much better thanseveral baseline methods.
The contribution of the paper is summarized as follows.
• We make the first step towards theoretically studying the loss design in PINN, and formallyintroduce the concept of stability in the literature of PDE to characterize the quality of alearned solution with small but non-zero Physics-Informed Loss.• We provide rigorous investigations on an important class of high-dimensional non-linearPDEs in optimal control, the HJB equation.
Our results suggest that the widely used L 2 lossis not a suitable choice for training PINN on HJB equations.• Based on the theoretical insight, we develop a novel PINN training algorithm to minimizethe L ∞ loss for HJB equations.
We empirically demonstrate that the proposed algorithmcan significant improve the accuracy of PINN in solving the optimal control problems.
Related Works
Physics-Informed Neural Network approaches [33, 28] learn to find parametric solutions to satisfyequations and boundary conditions with gradient descent.
There has been a notable scarcity ofpapers that rigorously justify why PINNs work.
Important works include [31], which prove theconvergence of PINN for second-order elliptic and parabolic equations.
In [21], the authors study thestatistical limit of learning a PDE solution from sampled observations for elliptic equations.
In [29],the convergence of PINN with L 2 loss is established for Navier-Stokes equations.
At the same time,several works observed different failure modes for training PINN in other PDE problems.
In [17],researchers discover that PINN sometimes fails to learn accurate solutions to a class of convectionand reaction equations.
Their analysis shows that this may be attributed to the complicated losslandscape.
[34] observed PINN failed to learn the Helmholtz equation due to the incommensurabilitybetween PDE and boundary losses.
In this work, we mainly experiment with the Hamilton-Jacobi-Bellman (HJB) equation in optimalcontrol.
Previously, there were several works aiming at solving the HJB equation using deep learningmethods [12, 13, 25, 39, 24, 1, 27, 5].
[13] is among the first to leverage neural networks to solve
HJB equations.
In particular, [13] targets constructing an approximation to a solution value u at
T = 0 , which is further transformed into a backward stochastic differential equation and learnedby neural networks.
The main difference between [13] and ours is that [13] only learns the solutionon a pre-defined time frame, while with our method, the obtained solution can be evaluated for anytime frame.
Recently, [12] tackled the offline reinforcement learning problem and developed a soft relaxation of the classical HJB equation, which can be learned using offline behavior data.
The maindifference between [12] and ours is that no additional data is required in our setting.
Stability is one of the most fundamental concepts in studying the well-posedness of PDE problems.
Formally speaking, it characterizes the behavior of the solution to a PDE problem when a smallperturbation modifies the operator, initial condition, boundary condition, or force term.
We saythe equation is stable if the solution of the perturbed PDE converges to the exact solution as theperturbations approach zero [8].
The problem regarding whether a PDE is stable has been intensivelystudied [8, 19, 11] in literature.
There are also some works studying how (regularity) conditionsaffect stability.
[20] and [7] investigate in which topology Couette Flow is asymptotic stable.
[4]answers in which Sobolev space defocusing nonlinear Schrodinger equation, real Korteweg-de Vries(KdV), and modified KdV are locally well-posed.
Our main focus is akin to the latter works but issettled in the machine learning framework.
Preliminary
In this section, we introduce basic background on Physics-Informed Neural Networks and stochasticoptimal control problems.
Without loss of generality, we formulate any partial differential equationas:
[Equation]
where L is the partial differential operator and B is the boundary condition.
We use x to denote thespatiotemporal-dependent variable, and use Ω and ∂ Ω to denote the domain and boundary.
Physics-Informed Neural Networks (PINN) PINN [28] is a popular choice to learn the functionu ( x ) automatically by minimizing the loss function induced by the PDE (1).
To be concrete, givenp ∈ (1 , + ∞ ) , we define the L p Physics-Informed Loss as
[Equation]
The loss term ℓ Ω ,p ( u ) in Eq.
(2) corresponds to the PDE residual, which evaluates how u ( x ) fitsthe partial differential equation on Ω ; and ℓ ∂ Ω ,p ( u ) in Eq.
(3) corresponds to the boundary residual,which measures how well u ( x ) satisfies the boundary condition on ∂ Ω .
L p denotes p -norm, where pis usually set to 2, leading to a “mean squared error” interpretation of the loss function [33, 28].
Thegoal is to find u ∗ that minimizes a linear combination of the two losses defined above.
The functionu ( x ) is usually parameterized by neural network u θ ( x ) with parameter θ ∈ Θ .
To find θ ∗ efficiently,
PINN approaches use gradient-based optimization methods.
Note that computing the loss involvesintegrals over Ω and ∂ Ω .
Thus, Monte Carlo methods are commonly used to approximate ℓ Ω ,p ( u )and ℓ ∂ Ω ,p ( u ) in practice.
Stochastic Optimal Control Stochastic control [9, 3] is an important sub-field in optimal controltheory.
In stochastic control, the state function { X t } 0 ≤ t ≤ T is a stochastic process, where T is thetime horizon of the control problem.
The evolution of the state function is governed by the followingstochastic differential equation:
[Equation]
where m : [ t, T ] × R n → R n is the control function and { W s } is a standard n -dimensional Brownianmotion.
Given a control function m , its total cost is defined as J x,t ( m ) =
E (cid:82) T t r ( X s , m, s )
d s + g
( X T ) ,where r : R n × R n × [0 , T ] → R measures the cost rate during the process and g : R n → R measuresthe final cost at the terminal state.
The expectation is taken over the randomness of the trajectories.
We are interested in finding a control function that minimizes the total cost for a given initial state.
Formally speaking, we define the value function of the control problem (4) as u ( x, t ) =
min m ∈M J x,t ( m ) ,where M denotes the set of possible control functions that we take into consideration.
It can beobtained that the value function will follow a particular partial differential equation as stated below.
Definition 3.1 ([38]) .
The value function u ( x, t ) is the unique solution to the following partialdifferential equation, which is called Hamilton-Jacobi-Bellman Equation :
[Equation]
Hamilton-Jacobi-Bellman (HJB) equation establishes a necessary and sufficient condition for acontrol’s optimality with regard to the cost functions.
It is one of the most important high-dimensional
PDEs [16] in optimal control with tremendous applications in physics [32], biology [18], and finance[26].
Many well-known equations, including Riccati equation, Linear–Quadratic–Gaussian controlproblem [36], Merton’s portfolio problem [22] are special cases of HJB equation [37].
Conventionally, the solution to the HJB equation, i.e., the value function u ( x, t ) , can be computedusing dynamic programming [2].
However, the computational complexity of dynamic programmingwill grow exponentially with the dimension of state function.
Considering that the state functionin many applications is high-dimensional, solving such HJB equations is notoriously difficult inpractice using conventional solvers.
As neural networks have shown impressive power in learninghigh-dimensional functions, it’s natural to resort to neural-network-based approaches for solvinghigh-dimensional HJB equations.
Failure Mode of PINN on High-Dimensional Stochastic Optimal Control
Note that u ( x ) is the exact solution to the PDE (1) if and only if both loss terms ℓ Ω ,p ( u ) and ℓ ∂ Ω ,p ( u )are zero.
However, in practice, we usually can only obtain small but non-zero loss values dueto the randomness in the optimization procedure or the capacity of the neural network.
In suchcases, a natural question arises: whether a learned u ( x ) with a small loss will correspond to a goodapproximator to the exact solution u ∗ ( x ) ?
Such a property is highly related to the concept stability in
PDE literature, which can be defined as below in our learning scenario:
Definition 4.1.
Suppose Z 1 , Z 2 , Z 3 are three Banach spaces.
We say a PDE defined as Eq.
(1)is ( Z 1 , Z 2 , Z 3 ) -stable, if ∥ u ∗ ( x ) − u ( x ) ∥
Z
=
O ( ∥L u ( x ) − φ ( x ) ∥
Z 1 + ∥B u ( x ) − ψ ( x ) ∥
Z 2 ) as∥L u ( x ) − φ ( x ) ∥
Z 1 , ∥B u ( x ) − ψ ( x ) ∥
Z 2 → 0 for any function u .
By definition, if a PDE is ( L 2 (Ω) , L 2 ( ∂ Ω) , Z ) -stable with a suitable Banach space Z , we can minimize the widely used L 2 Physics-Informed Losses ∥L u ( x ) − φ ( x ) ∥ 2 L 2 (Ω) and ∥B u ( x ) − ψ ( x ) ∥ 2 L 2 ( ∂ Ω) ,and the learned solution is guaranteed to be close to the exact solution when the loss terms approachzero.
However, stability is not always an obvious property for PDEs.
There are tremendous equationsthat are unstable, such as the inverse heat equation.
Moreover, even if an equation is stable, it ispossible that the equation is not ( L 2 (Ω) , L 2 ( ∂ Ω) , Z ) -stable, which suggests that the original L
Physics-Informed Loss might not be a good choice for solving it.
We will show later that for controlproblems, some practical high-dimensional HJB equations are stable but not ( L 2 (Ω) , L 2 ( ∂ Ω) , Z ) -stable, and using L 2 Physics-Informed Loss will fail to find an approximated solution in practice.
We consider a class 1 of HJB equations in which the cost rate function is formulated as r ( x, m )
=a 1 | m 1 | α 1 + · · · + a n | m n | α n − φ ( x, t ) .
The corresponding Hamilton-Jacobi-Bellman equation canbe reformulated as:
[Equation]
where A i =
( a i α i ) − 1 αi − 1 − a i ( a i α i ) − αiαi − 1 ∈ (0 , + ∞ ) and c i = α
i α
i − 1 ∈ (1 , ∞ ) .
See Appendix Bfor the detailed derivation.
For a function f : X → R , where X is a measurable space, we denote bysupp f the support set of f , i.e. the closure of { x ∈ X : f ( x ) ̸ = 0 } .
An important concept for analyzing PDEs is the Sobolev space, which is defined as follows
Definition 4.2.
For m ∈ N , p ∈
[1 , + ∞ ) and an open set Ω ⊂ R n , the Sobolev space W m,p (Ω) isdefined as { f ( x ) ∈
L p (Ω) : D
α f ∈ L p (Ω) , ∀ α ∈ N n , | α | ≤ m } .
The function space W m,p (Ω) isequipped with Sobolev norm, which is defined as ∥ f ∥ W m,p (Ω) = (cid:32) (cid:80) |
[Equation]
|≤ m
[Equation]
D
[Equation]
f
[Equation]
pL p (Ω) (cid:33) 1 p .
The definition above can be extended to functions defined on a spatiotemporal domain Q ⊆ R n ×[0 , T ] .
With a slight abuse of notation, we define W m,p ( Q ) =
{ f ( x, t ) ∈
L p ( Q ) :
D α f ∈
L p ( Q ) , ∀ α ∈ N n , | α | ≤ m } , where the differential D α is only operated over spatial variable x .
Thenorm ∥ · ∥ W m,p ( Q ) can also be defined accordingly.
Stability of the HJB Equation We present our main theoretical result which characterizes thestability of the HJB equation (Eq.
In particular, we show that the HJB equation is ( L p ( R n ×[0 , T ]) , L q ( R n ) , W 1 ,r ( R n × [0 , T ]))
-stable when p , q and r satisfies certain conditions.
Theorem 4.3.
For p, q ≥ 1 , let r 0 =
( n +2)
q n + q .
Assume the following inequalities hold for p, q and r 0 :
[Equation]
where ¯ c = max 1 ≤ i ≤ n c i in Eq.
Then for any r ∈ [1 , r 0 ) and any bounded open set Q ⊂ R n × [0 , T ] ,
Eq.
(6) is ( L p ( R n × [0 , T ]) , L q ( R n ) , W 1 ,r ( Q ))
-stable for ¯ c ≤ 2 .
The proof of Theorem 4.3 can be found in Appendix C and an improved theorem with relaxeddependency on ¯ c can be found in Appendix E. Intuitively, Theorem 4.3 states that ( L p , L q , W 1 ,r ) -stability of Eq.
(6) can be achieved when p, q = Ω( n ) .
We further show that this linear dependencyon n cannot be relaxed in the following theorem:
Theorem 4.4.
There exists an instance of Eq.
(6), whose exact solution is u ∗ , such that for anyε > 0 , A > 0 , r ≥ 1 , m ∈ N and p ∈ (cid:2) 1 , n 4 (cid:3) , there exists a function u ∈ C ∞ ( R n × (0 , T ]) whichsatisfies the following conditions:
• ∥L HJB u − φ ∥
L p ( R n × [0 ,T ]) < ε , B
HJB u = B HJB u ∗ , and supp( u − u ∗ ) is compact, where
L HJB and B HJB are defined in Eq.
• ∥ u − u ∗ ∥ W m,r ( R n × [0 ,T ]) >
A.
The proof of Theorem 4.4 can be found in Appendix D.
Discussion.
Theorem 4.3 and 4.4 together state that when the dimension of the state function n islarge, the HJB equation in Eq.
(6) cannot be ( L p , L q , W 1 ,r ) -stable
if p and q are small.
Therefore,for high-dimensional HJB problems, if we use classic L 2 Physics-Informed Loss for training PINN,the learned solution may be arbitrarily distant from u ∗ even if the loss is very small.
Such theoreticalresults are verified in our empirical studies in Section 6.
More importantly, our theoretical results indicate that the design choice of the Physics-Informed Lossplays a significant role in solving PDEs using PINN.
In this work, we shed light upon this problemusing HJB equations.
We believe the relationship between PDE’s stability and the Physics-Informed
Loss should be carefully investigated in the future, especially for high-dimensional non-linear PDEswhose stability are more complicated than low-dimensional and linear ones [8, 11, 19].
Given theabove observations, we further propose a new algorithm for training PINN to solve HJB Equations,which will be presented in the subsequent sections.
Solving HJB Equations with Adversarial Training
The above results suggest that we should use a large value of p and q in the loss ℓ Ω ,p ( u ) and ℓ ∂ Ω ,q ( u )to guarantee a learned solution u is close to u ∗ for high-dimensional HJB problems.
Note that
L p -norm and L ∞ -norm behave similarly when p is large.
We can substitute L p -norm by L ∞ -normand directly optimize ℓ Ω , ∞ ( u ) and ℓ ∂ Ω , ∞ ( u ) .
Overall, the training objective can be formulated as:
[Equation]
Input: Target PDE (Eq.
(1)); neural network u θ ; initial model parameters θ
Output: Learned PDE solution u θ
Hyper-parameters: Number of total training iterations M ; number of iterations and step size of inner loop K, η ; weight for combining the two loss term λ 1: for i = 1 , · · · , M do 2:
Sample x (1) , · · · , x ( N 1 ) ∈ Ω and ˜ x
(1) , · · · , ˜ x ( N 2 ) ∈
∂
Ω 3: for j = 1 , · · · , K do 4: for k = 1 , · · · , N 1 do 5: x ( k ) ← Project Ω (cid:16) x ( k ) + η sign ∇ x (cid:0) L u θ ( x ( k ) ) − φ ( x ( k ) )
(cid:1) 2 (cid:17) 6: for k
= 1 , · · · , N 2 do 7:
˜ x ( k ) ← Project ∂ Ω (cid:16) ˜ x ( k ) + η sign ∇ x (cid:0) B u θ (˜ x ( k ) ) − ψ (˜ x ( k ) ) (cid:1) 2 (cid:17) 8: g ← ∇ θ (cid:32)
N
N 1 (cid:88) i = (cid:16) L u θ ( x ( i ) ) − φ ( x ( i ) )
(cid:17) 2 + λ · 1 N
N 2 (cid:88) i = (cid:16) B u θ (˜ x ( i ) )
− ψ (˜ x ( i ) )
(cid:17)(cid:33) 9: θ ← Optimizer ( θ, g ) 10: return u θ where λ > 0 is a hyper-parameter to trade off the two objectives.
It is straightforward to obtain that setting p and q to infinity satisfies the conditions in Theorem 4.3,and thus the quality of the learned solution enjoys theoretical guarantee.
Furthermore, Eq.
(8) can beregarded as a min-max optimization problem.
The inner loop is a maximization problem to find datapoints on Ω and ∂ Ω where u violates the PDE most, and the outer loop is a minimization problem tofind u (i.e., the neural network parameters) that minimizes the loss on those points.
In deep learning, such a min-max optimization problem has been intensively studied, and adversarialtraining is one of the most effective learning approaches in many applications.
We leverage adversarialtraining, and the detailed implementation is described in Algorithm 1.
In each training step, the modelparameters and data points are iteratively updated.
We first fix the model u and randomly sampledata points x (1) , · · · , x ( N 1 ) ∈ Ω and ˜ x (1) , · · · , ˜ x ( N 2 ) ∈
∂ Ω , serving as a random initialization of theinner loop optimization.
Then we perform gradient-based methods to obtain data points with largepoint-wise Physics-Informed Losses, which leads to the following inner-loop update rule:
[Equation]
where Project Ω ( · ) and Project ∂ Ω ( · ) project the updated data points to the domain.
When theinner-loop optimization finishes, we fix the generated data points and calculate the gradient g to themodel parameter:
[Equation]
then the model parameter can be updated using any first-order optimization methods.
When thetraining finishes, the learned neural network will converge to a solution with small L ∞ losses and isguaranteed to be close to the exact solution.
Experiments
In this section, we conduct experiments to verify the effectiveness of our approach.
Ablation studieson the design choices and hyper-parameters are then provided.
Our codes are implemented basedon PyTorch [23].
All the models are trained on one NVIDIA Tesla V100 GPU with 16GB memory.
Codes can be found in the supplementary material.
Due to space limitation, we only showcase our
[Table]
methods on the Linear Quadratic Gaussian control problem in the main body of the paper.
Moreexperimental results on other PDE problems can be found in the appendix.
6.1 High Dimensional Linear Quadratic Gaussian Control Problem
We follow [13] to study the classical linear-quadratic Gaussian (LQG) control problem in n dimensions, a special case of the HJB equation, defined as below:
[Equation]
As shown in [13]
[Table]
there is a unique solution to Eq.
[Equation]
We set µ = 1 , T = 1 , and the terminal cost function g ( x ) =
ln (cid:18) 1 + ∥ x
∥
[Equation]
Experimental Design
The neural network used for training is a 5-layer MLP with 4096 neuronsand tanh activation in each hidden layer.
To train the models, we use Adam as the optimizer [15].
The learning rate is set to 3e − 4 in the beginning and then decays linearly to zero during training.
Thetotal number of training iterations is set to 5000/10000 for the 100/250-dimensional problem.
In eachtraining iteration, we sample N 1 = 100 / 50 points from the domain R n × [0 , T ] and N 2 = 100 / 50points from the boundary R n × { T } to obtain a mini-batch for the 100/250-dimensional problem.
The number of inner-loop iterations K is set to 20, and the inner-loop step size η is set to 0.05 unlessotherwise specified.
Evaluations are performed on a hold-out validation set which is unseen duringtraining.
We use the L 1 relative error in the domain [0 , 1] n × [0 , T ] and the L 1 relative error in theboundary [0 , 1] n × { T } as evaluation metrics to quantify how well the model fits the exact solutionin the domain and over the boundary, and we refer to these two metrics as the “domain relative error”and “boundary relative error” respectively.
More detailed descriptions of the experimental setting canbe found in Appendix F.
We compare our method with a few strong baselines: 1) original PINN trained with L 2 PhysicsInformed Loss [28]; 2) adaptive time sampling for PINN training proposed in [35]; 3) PINN withthe learning rate annealing algorithm proposed in [34]; 4) curriculum PINN regularization proposedin [17].
The training recipes for the baseline methods, including the neural network architecture,the training iterations, the optimizer in the learning rate, are the same as those of the our methoddescribed above.
It should be noted that although these approaches modifies the data sampler, trainingalgorithms or the loss function, they all keep the L 2 norm of the PDE residual and boundary residualunchanged in the training objective.
Experimental Results
The experimental results are summarized in Table 1.
It’s clear that therelative error of the model trained using the original PINN does not fit the solution well, e.g., the
[Figure]
domain relative error is larger than 5% when n = 250 .
This empirical observation aligns well withour theoretical analysis, i.e., minimizing L 2 loss cannot guarantee the learned solution to be accurate.
Advanced methods, e.g., curriculum PINN regularization [17], can improve the accuracy of thelearned solution but with marginal improvement, which suggests that these methods do not addressthe key limitation of PINN in solving high-dimensional HJB Equations.
By contrast, our proposedmethod learns accurate solutions and significantly outperforms all the baseline methods.
We also examine the quality of the learned solution u ( x, t ) by visualization.
As the solution is ahigh-dimensional function, we visualize its snapshot on a two-dimensional space.
Specifically, weconsider a bivariate function u ( x 1 , x 2 , 0 , · · · , 0; 0) and use a heatmap to show its function valuegiven different x 1 and x 2 .
Figure 1 shows the ground truth u ∗ , the learned solutions u of original
PINN and our method, and the point-wise absolute error | u − u ∗ | for each methods.
The two axisescorrespond x 1 and x 2 , respectively.
We can see that the point-wise error of the learned solutionusing our algorithm is less than 2e − 2 on average.
In contrast, the point-wise error of the learnedsolution using original PINN method with L 2 loss is larger than 1 .
3e − 1 for most areas.
Therefore,the visualization of the solutions clearly illustrate that PINN learned more accurate solution using ourproposed algorithm.
6.2 Ablation studies
We conduct ablation studies on the 100-dimensional LQG control problem (Eq.
(12)) to ablate themain designs in our algorithm.
Adversarial training v.s. Directly optimizing L p physic-informed loss.
Our proposed algorithmintroduces a min-max optimization procedure.
One may have concerns that such an approach may beunnecessarily complicated, and directly minimizing L p physic-informed loss with a large p wouldhave the same effect.
We use these two methods to solve Eq.
(12), and compare their performance inthe left panel of Table 2.
It can be seen that directly minimizing L p physic-informed loss does notlead to satisfactory results.
For detailed discussions, please refer to Appendix H.
Adversarial training should be applied to both the PDE residual and the boundary residual.
Our theoretical analysis suggests that we should use a large value of p and q in the loss ℓ Ω ,p ( u )and ℓ ∂ Ω ,q ( u ) to guarantee the quality of the learned solution u .
Thus, in the proposed Algorithm1, both the data points inside the domain and the data points on the boundary are learned in the
[Table]
inner-loop maximization.
From the right panel of Table 2, we can see that when adversarial training isapplied to one loss term, the performance is slightly improved, but its accuracy is still not satisfactory.
When both loss terms use adversarial training, the solution is one order of magnitude more accurate,indicating that applying adversarial training to the whole loss function is essential.
Hyper-parameters K and η for the inner loop maximization.
Our approach introduces additionalhyper-parameters K (the number of inner-loop iterations) and η (the inner-loop step size).
Thesetwo parameters control the accuracy of inner-loop maximization.
We conduct ablation studies toexamine the effects of different design choices.
Specifically, we experiment with K = 5 , 10 , 20 andη = 0 .
05 , 0 . 1 , 0 . 2 , and show the domain error and boundary error in Figure 2.
Typically we find thatsetting the product Kη = 1 achieves the best performance.
When Kη is fixed, our results suggestthat using a larger K and a smaller η , i.e., more inner-loop iterations and smaller step sizes, will leadto better performance while being more time-consuming.
Conclusions
In this paper, we theoretically investigate the relationship between the loss function and the approximation quality of the learned solution using the concept of stability in the partial differentialequation.
We study an important class of high-dimensional non-linear PDEs in optimal control, the
Hamilton-Jacobi-Bellman (HJB) equation, and prove that for general L p Physics-Informed Loss, the
HJB equation is stable only if p is sufficiently large.
Such a theoretical finding reveals that the widelyused L 2 loss is not suitable for training PINN on high-dimensional HJB equations, while L ∞ loss isa better choice.
The theory also inspires us to develop a novel PINN training algorithm to minimizethe L ∞ loss for HJB equations in a similar spirit to adversarial training.
One limitation of this workis that we only work on the HJB Equation.
Theoretical investigation of other important equations canbe an exciting direction for future works.
We believe this work provides important insights into theloss design in Physics-Informed deep learning.
Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
[Yes]
(b) Did you describe the limitations of your work?
[N/A](c)
Did you discuss any potential negative societal impacts of your work?
[No] We don’tsee any potential negative societal impacts of this work.(d)
Have you read the ethics review guidelines and ensured that your paper conforms tothem?
[Yes]2.
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes]
(b) Did you include complete proofs of all theoretical results?
[Yes] The proofs areprovided in the appendix.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or as a URL)?
[Yes] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
[Yes](c)
Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
[N/A](d)
Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes]4.
If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[Yes]
(b) Did you mention the license of the assets?
[Yes]
(c) Did you include any new assets either in the supplemental material or as a URL?
[No]
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[N/A]
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount
spent on participant compensation?
[N/A]