Meta-Auto-Decoder for Solving Parametric Partial Differential Equations
Authors: Xiang Huang
Zhanhong Ye
Hongsheng Liu
Shi Bei Ji
Zidong Wang
Kang Yang
Yang Li
Min Wang
Haotian CHU
Fan Yu
Bei Hua
Lei Chen
Bin Dong
Venue: NeurIPS 2022 Accept (2022)
Keywords: Parametric PDE
Meta-learning
Auto-decoder
Physics-Informed Learning

Abstract: Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.  Recently, building learning-based numerical solvers for parametric PDEs has become an emerging new field.  One category of methods such as the Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs) aim to approximate the solution of the PDEs. They are typically unsupervised and mesh-free, but require going through the time-consuming network training process from scratch for each set of parameters of the PDE.  Another category of methods such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) try to approximate the solution mapping directly.  Being fast with only one forward inference for each PDE parameter without retraining, they often require a large corpus of paired input-output observations drawn from numerical simulations, and most of them need a predefined mesh as well.  In this paper, we propose Meta-Auto-Decoder (MAD), a mesh-free and unsupervised deep learning method that enables the pre-trained model to be quickly adapted to equation instances by implicitly encoding (possibly heterogenous) PDE parameters as latent vectors.  The proposed method MAD can be interpreted by manifold learning in infinite-dimensional spaces, granting it a geometric insight.  Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods.

Meta-Auto-Decoder for Solving Parametric Partial Differential Equations
Xiang Huang"
Zhanhong Ye" yezhanhong@pku. edu. cn Peking University
Hongsheng Liu Liubongsheng4@huuxel com Huawei Technologies Co. Ltd
sahx@mail ustc. edu. University of Science and Technology of China
Beiji Shi Chibeiji@huarae com Huawei Technologies Co. Ltd
Zidong Wang wang1 mang1hhuwwe..cmm com Huawei Technologies Co. Ltd
Kang Yang yangkang22@huaveii com Huawei Technologies Co. Ltd
Yang Li 1iyang477@huaa wwwwwei. com com Huawei Technologies Co. Ltd
Min Wang vangmin106@huare com Huawei Technologies Co. Ltd
Haotian Chu chuhaot iaa.ottii@@@hhaeeei com Huawei Technologies Co. Ltd
Fan Yu fan. yu@huawei com Huawei Technologies Co. Ltd
Bei Hua bhua@ustc. edu. cn University Science and Technology of China
Lei Chen
leicheecse ust hk Hong Kong University of Science and Technology
Bin Dong
dongbinGmath. pku.peeeee edu. cn Beijing International Center for Mathematical Research, Peking University Center for Machine Learning Research, Peking University
Abstract
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc. Recently, building learning-based numerical solvers for parametric PDEs has become emerging new field. One category of methods such as the Deep Galerkin Method (DGM) and Physics-Inforee Informed Neural Networks (PINNs) aim to approximate the solution of the PDEs. They are typically unsupervised and feesh-free, but require going through the consumnnn network training process from scratch for each set of parameters of the PDE. Another category of methods such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) try approximate the solution mapping directly. Being fast with only one forward inference for each PDE parameter without retraining. they often require large corpus of paired input-output observations drawn from numerical simulations. and most of them need a predefined mesh as well. this paper, we propose Meta-Auto- Decoder (MAD), mesh-free and unsupervised deep learning method that enables
'The first authors contributed equally this paper, and Bin Dong the corresponding author. Zhanhong Ye proposed MAD-L and explain the effectiveness of MAD method from the perspective of manifold learning. Huang Xiang proposed MAD-LM on the basis of MAD-L and completed all the experiments in the paper. Xiang Huang performed this work during an internship at Huawei.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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the pre-trained model to be quickly adapted to equation instances by implicitly encoding (possibly heterogenous) PDE parameters as latent vectors. The proposed method MAD can be interpreted by manifold learning in infiite-dimensional spaces, granting a geometric insight. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods. The project page with code is available: http:////..tottooooooo/////////s
1 Introduction
Many important problems in science and engineering, such as inverse problems, control and optimiza- tion, risk assessment, and uncertainty quantification (I12), require solving the so-called parametric PDEs, i.e., partial differential equations (PDEs) with different physical parameters, boundary condi- tions, or solution regions. Mathematically, they require solve the so-called parametric PDEs that can be formulated as:
vu= E EENCR
BT = 0, I E an
(1)
where 2 and T are partial differential operators parametrized by "n and 72. respectively, and denotes the independent variable in spatiotemporrl-deeeedeee PDEs. Given U ((2; Rd)) and the space of parameters A, = (71, "/2, S2) E A is the variable parameter of the PDEs and u E U is the solution of the PDEs. Note that the form of 17 considered here is very general with possible heterogeneity allowed, since the computational domain shape Z and the functions defined on this domain its boundary (which may be involved in 7i, 72) is obviously of different type. Solving parametric PDEs requires to learn an infinite-dimensiona operator G A - Ui that map any PDE parameter 7) to its corresponding solution u" (i.e., the solution mapping).
In recent years, learning-basse based PDE solvers have become very popular, and it is generally believed that learning-basee -bsend PDE solvers have the potential to improve efficiency [3] 4 []. The learning- based PDE solvers can be categorized into two categories in terms of the objects that are approximated by neural networks (NN), i.e., the approximation of the solution u²" and the approximation of the solution mapping G.
NN as new ansatz of solution. This kind of approaches approximate the solution of the PDEs with a neural network and mainly rely on governing equations and boundary conditions (or their variants) train the neural networks. For example, PINNs (3] and DGM (6) constrain the output of deep neural networks to satisfy the given governing equations and boundary conditions. Deep Ritz Method (DRM) 7 exploits the variational form of PDEs and can be used to solve PDEs that can be reformulated as equivalent energy minimization problems. Based on weak formulation of PDEs, Weak Adversarial Network (WAN) & parameterizes the weak solution and test functions as primal and adversarial neural networks, respectively. These neural approximation methods can work in an unsupervised manner, without the need to generate labeled data from conventional computational methods. However, all these methods treat different PDE parameters independent tasks, and need to retrain the neural network from scratch for each PDE parameter. When a large number of tasks with different PDE parameters need to be solved, these methods are computationally expensive and impractical. In order to mitigate retraining cost, E and Yu (7) recommends a transfer learning method that uses model trained for one task the initial model to train another task. However, according to our experiments, transfer learning method is not always effective in improving convergence speed (see Sec 3.1 3.3
NN as a new ansatz of solution mapping, This kind of approaches use neural networks to learn the solution mapping between two infinitediimensional function spaces (9 IO [T 12 [3]. For example, PDE-Nets [9 [0] are among the earliest neural operators that are specifically designed convolutional neural networks inspired by finite difference approximations of PDEs. They are able to uncover hidden PDE models from observed dynamical data and perform fast and accurate predictions at the same time. DeepONet [ uses two subnets to encode the parameters and location variables of the PDEs separately, and merge them together to compute the solution. FNO [33 utilizes fast
Fourier transform to build the neural operator architecture and learn the solution mapping between two dimeite-dimensionnll function spaces. A significant advantage of these approaches is that once the

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neural network is trained, the prediction time is almost negligible. Although they have demonstrated promising results across a wide range of applications, several issues occur. First, the data acquisition cost is prohibitive in complex physical, biological, or engineering systems, and the generalization ability of these models is poor when there is not enough labeled data [1]. Second, most of these methods p 10, 12 [3] require predefined mesh and utilize the labeled data on the mesh for training and inference. Third, simply applying one forward inference may lead to unsatisfactory generalization, especially on out-of-ditrrbution (OOD) settings (i.e., PDE parameters for training and inference are from different probability distributions). Finally, these operators directly takes the PDE parameter 17 as network imput, which would bring inconvenience in network implementation if n is heterogeneous. The recently proposed Physics-Informel DeepONet (PI-DeepONet) [15] can learn mesh-free solution mapping without any labeled data and retraining. However, it needs to collect a large number of training samples in the parameter space A to obtain an acceptable accuracy (see and is still inflexible dealing with heterogeneous PDE parameters.
Sec 3.1
Meta-Learning. Different from conventional machine learning that learns to do given task, meta-learning learns improve the learning algorithm itself based on multiple learning episodes over distribution of related tasks. As a result, meta-learning can handle new tasks faster and better. In this field, the Model-Agnostic Meta-Learning (MAML) (16) algorithm and its variants [7 18 [9] have beed widely used. These algorithms try to find an initial model with good generalization ability such that it can be adapted new tasks with small number of gradient updates. For example, MAML [16 firstly trains a meta- model with good initialization weight on a variety of learning tasks, which is then fine-tuned on a new task through a few steps of gradient descent to get the target model. The Reptile [18] algorithm eliminates second-order derivatives in MAML algorithm by repeatedly sampling a task, training on it, and moving the initialization towards the trained weight on that task.
Borrowing the idea of meta-learning may inspire new ways to solve parametric PDEs, where different PDE parameters correspond different tasks. To the best of our knowledge, Meta-MgNet [20] is the first work that view solving parametric PDEs as a meta-learning problem, which is based on hypernet and the multigrid algorithm. Meta-MgNet utilizes the similarity between tasks to generate good smoothing operators adaptively, and thereby accelerates the solution process, but is not directly applicable to PDEs on which the multigrid algorithm is not available. Recently, the Reptile algorithm is also used to accelerate the PDE solving problems in [21). However, MAML and Reptile are not always effective in improving convergence speed (see Sec and 3.3).
Our contributions. We propose Meta- Auto-Atttooooo- (MAD), a mesh-ffree and unsupervised deep learning method that enables the pre-trained model to be quickly adapted to equation instances by implicitly encoding heterogeneous PDE parameters as latent vectors. Different from Meta-MgNN MAD makes use of the similarity between tasks from the perspective of manifold learning, and tries to learn nonlinear approximation of the solution manifold. We construct the ansatz of solution as neural network in the form us (T, 2). By taking the spatial (or ppatial-temporall coordinate T directly as the network input, unsupervised training loss is allowed, and mesh is no longer required. As the additional input = varies, up un 2) moves on a manifold in an infinite-dimensione function space, which may be approximation of the true solution manifold for certain 0. The PDE parameter v) is implicitly encoded into z by applying the auto-decoder architecture motivated by 22 regardless of the possible heterogeneity. When a new task comes, MAD achieves fast transfer by projecting the new task to the manifold and fine-tuning the manifold at the same time. The main contributions of this paper are summarized as follows:
A mesh-free and unsupervised deep neural network approach is proposed to solve parametric PDEs. Based on meta-learning concept, once the neural network is pre-trained, solving a new task involves only a small number of iterations. In addition, the auto-decoder architecture adopted by MAD can realize auto-encoding of heterogeneous PDE parameters.
The mathematical intuition behind the MAD method is analyzed from the perspective of manifold learning. In short, a neural network is pre-trained to approximate the solution manifold, and the required solution is searched on the solution manifold or in neighborhood of the solution manifold.
Extensive numerical experiments are carried out to demonstrate the effectiveness of our method, which show that MAD can significantly improve the convergence speed and has good extrapolation ability for OOD settings.
3

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PDE(11)
implicitly
C U
(thy 7a, !
Loss
1-
BC(22)
Nearal Network: dip
Figure 1: Architecture of Meta-Auto-Decoderr
2 Methodology
2.1 Meta-Auto-Deccoder
We adopt meta-learning concept realize fast solution of parametric PDEs. Our basic idea is first learn some universal meta-knowledge from set of sampled tasks in the pre-training stage, and then solve new task quickly by combining the task-specific knowledge with the shared meta-knowledge in the fine-tuning stage. We also adapt the auto-decoder architecture in [22], and introduce uy ( =) approximate the solutions of parametric PDEs. The architecture of u 2) is shown Fig thysics-informer loss is used for training, making the proposed method unsupervised. Putting all these together, we propose new method Meta-Aute Aecoder (MAD) solve parametric PDEs. For the rest of the subsection, the loss function and the two stages of training will be explained in details.
To enable unsupervised learning, given any PDE parameter 1) A, the physics-informer loss L u + [0, oo) about Eq.[
L' || u Abe |B w (22
is considered, where Ane > 0 is weighting coefficient. The Monte Carlo estimate of L*
M.
Mr
ABe
[ =
Σ M,
M
2
3=1
where (K) (. and { (I......,MMe) are two sets of random sampling points in Ω and 8N, respectively. This task-specific loss ['[u] can be computed by automatic diffcrentiation (2 and will be used in the pre-training stage and the fine-tuing stage.
In the pre-training stage, through minimizing the loss function, a pre-trained model parametrized by 8" is learned for all tasks and each task is paired with its own decoded latent vector 2 Such pre-trained model is considered as meta knowledge as is learned from the distribution of tasks and the learned latent vector 2, is the task-specific knowledge. When solving new task in latent may unfreeze 0 and allow it to be fine-tuned along with These two fine-tuning strategies give rise to different versions of MAD, which are called MAD-L and MAD-LA LM, respectively. The corresponding problems of ree-training and fine-tuning are formulated as follows:
fine-tuning stage, keep the model weight 0* fixed and minimize the loss by fine-tuning vector Alternatively,
.IN E A. both MAD-L Given N randomly generated PDE parameters ",...., and MAD-IM solve the following optimization problem
Pre-training Stage
(in(uo() (x)) @ 1=:113)
((=))ee(,...,N) 0-)
arg min 0.
(4)
where 0* is the optimal model weight, {2, (e.......)) (1. are the optimal latent vectors for different PDE parameters, and Lhh is defined in Eq. B.. The regularization _/2 +|/2||1 is added for training stability.

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Fine-tuning Stage (MAD-L) Given a new PDE parameter Mnww, MAD-L keeps 6" fixed, and minimizes the following loss function to get
arg min i. Neww (uo- (1,2)] + 0² 11=11?
(5)
new
then ugg (t, ((ew) is the approximate solution of PDEs with parameter 7hew- To speed up convergence, we can set the initial value of z to 27 obtained during pre-training where 17, is the neares to neww-
Fine-tuning Stage (MAD-LM) MAD-LM fine-tunes the model weight 0 with the latent vector simultaneously, and solves the following optimization problem
nZno (new) = arg min i T (ua( =)] + क2 |=
2.0
with initial model weight H* This would produce an alternative approximate solution ug; (1 new ). The latent vector is initialized in the same way as MAD-L.
Remark The MAD method has several key advantages compared with existing methods. Besides being mesh-free and unsupervised. it can deal with heterogeneous PDE parameters painlessly, since n is not taken as the network input, and is encoded into z in an implicit way. Introduction of the meta- kntwledge 6* would accelerate the fine-tunint process, which can be better understood in the light of the manifold learning perspective. For MAD-LM, the accuracy on OOD tasks likely to be at least comparable with training from scratch based on PINNs. Although the fine-tuning process of MAD is still slower than one forward inference of neural network solution mapping, the advantages presented above can make it more suitable for some real applications.
If we replace the would then coincide with the DeepSDF algorithm [22]. Despite this, the field of solving parametric PDEs is quite different from 3D shape representation in computer graphics. Moreover, the introduction of model weight tme-tuming in MAD-LM can significantly improve solution accuracy, as is explained
Remark 2
loss by cerluin supervised loss, the MAD-L method
intuitively in Sec 2.2 2.3 and validated by numerical experiments in Sec
2.2 Manifold Learning Interpretation of MAD-I
We interpret how the MAD-L. method works from the manifold learning perspective, which also provides new interpretation of the DeepSDF algorithm (2]. For the rest of this section, the domain Ω is fixed and excluded from for simplicity. Now, we consider the following scenario.
The set of solutions G(AA = (G(n) I E A} C is contained in a low-dimensional structure. To be more specific, there is a fimteediimnsoo space Z = R (with & dimld and Lipschitz continuous mapping G Z + U, such that G(A) C G(Z). In other words, for any E A.
Scenario 1
there exists E satisfying G(z) G(7).
The mapping G is Lipschitz continuous if and only there exists some C such that [|G(z) G(z')] u < ClI== - =²'l for all 2, , E Z. This Lipschitz continuous constraint excludes highly irregular mappings like space-filling curves. When A is a finitediimensional space and G is Lipschitz continuous, the parametric PDE would fall into this scenario (just take Z A, G = G). Since dim Z < dimld (the latter is usually infinity) holds, we may view the mapping G as some sort of "decoder", and Z is the corresponding latent vector space, despite of the fact that there doesn't 't exist an "encoder". In many cases, G(Z)= Z forms embedded submanifold, and therefore our MAD method can be viewed as a manifold-learning approach. Once the mapping & is learned as above, then for a given parameter 77, searching for the solution u" in the whole space 2/ is no longer needed. Instead, we may focus on the smaller subset G (Z), i.e. the class of functions in 2/ that is parametrized by Z, since n G(n) G(Z) holds for any 1) E A. We then solve the optimization problem
arg min [=))),
For example, if A is a space of functions, we can discretize function into a vector and then find the Euclidean distance between the two vectors as the distance between two PDE parameters.
5

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0.4)
f4 fr 2))
given Thens we fine-tune
for (1)
fa 1.-
Jv m2)
fine-tune
U
line tune simultanepuslyy
(a) Fine-tune only (MAD-L)
(b) Fine-tune both = and 0 (MAD-LM)
Illustration of how MAD works from the manifold learning perspective. (a) MAD-L: The function space V is mapped to a 2-dimensional plane. The blue solid curve represents the solution set G(A) formed by exact solutions corresponding all possible PDE parameters, and each point on the curve represents an exact solution corresponding to one PDE parameter. The orange dotted curve represents the solution set Gg- (Z) obtained by the pre-trained model, and each point on the curve corresponds to latent vector Given Tnee E A., rather than searching in the entire function space Ut MAD-L only searches on the orange dotted curve find an optimal æ such that its corresponding solution ug. (+)z) is nearest to the blue point uhe.. (b) MAD-LM: The solution set ((A) lies within a neighborhood of Gg. (Z) that is represented by a gray shadow band. To find solution u²h",, we have to fine-tune 9 (i.e., the orange dotted lines) and the latent vector 2 (i.e., the points on the orange dotted lines) simultaneously to approach the exact solution. As the search scope is limited a strip with small width, the fine-tuning process can be expected to converge quickly.
Figure 2:
and (22") the approximate solution. Assuming that the dimension of is chosen (either empirically or through trial and error), the aim is to find the mapping G. Since such mapping is usually complex and hard to design by hand, we consider the 0-parametrized version Gy Z + U, and find the best 0 automatically by solving an optimization problem. Gg can be constructed in the simple form
Gz(z)(z) us ()),
where Ue is a neural network whose input is the concatenation of ă E Rd and 2 E RR, The next step is to find the optimal model weight 0 via training, with the target being G(A) S Gz(Z). Assuming that the PDE parameters are generated from a probability distribution 17 ~ P.A. then G(n) E Go(Z) holds almost surely if and only if
d(0)
E ツ~PA
[uu("", (T(,))
minn|uu uo( MMx 0,
which suggests taking 0- = arg ming d(O). In case we do not have direct access to the exact solutions w", the equivalent" condition
conditiio
d' (0)
E PoPP
min T." [ =)]
(10)
is considered, and d' (8) becomes the alternative loss to be minimized.In the specific implementation, the expectation on 1) ~ P.a is estimated by Monte Carlo samples 1,,..., n/, and the optimal network weight 0 is taken to be
8" = arg min 9
1 N
Σ s=1
min th (.
(11)
We further estimate the physics-iffrrmed loss L using Monte Carlo method to obtain Eq.4 After
that, when new PDE parameter 7hew A comes, direct adaptation of Eq.(7) would then give rise to the fine-tuning process Eq. (5). since tig. (.,2) Gø- (2) Rs G(z) holds. An intuitive illustration of how MAD-L works from the manifold learning perspective is given in Fig 2(a)
Two types of parametrization considered here. The latent vector parametrizes point the manifold G(Z) or GZ) and parametrizes the shape of the entire manifold Go (2). "Assume that the solution of Eq. (T] is unique for all E A, and E 24 is the solution if and only if L L"[u] 0.

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G(A)
G(A)
Gym (2)
Gg. (2)
Gym
G(thee)
Gyzn (23
Gy (Shne)
Gy (Fack)
(Gomita/ (0
X
G (2)
[Gpm (M /o (1...s)
G (Ppmk
[Goom ( )).e (100))))
G. (Pom))
2.5
-3.0
-25
5.0
2.5
5.0
(a) Pre-training
(b) Fine-tuning
Figure 3: Visualization of the MAD pre-training and fine-tuning process for the ODE problem.
Visualization Example An ordinary differential equation (ODE) is used to visualize the pre- training and fine-tuning processes of MAD-L. Consider the following problem with domain (- π C R:
du
- 11) cos ((x n));),
u( u(+)) sin ((+ -))²).
(12)
dr
We sample 20 points equidistantly the interval [0, 2] variable ODE parameters, and randomly select one Vhew for fine-tuning stage and the rest (7) (i(,,.-.1))) for pre-training stage. MAD-I.
generates a sequence of (8(m), {2, (m))) (191).-.199)) in pre-training stage, and terminates at m 200
with the optimal (0", {="] (e.......... ..))). The infinit-diimensionall function space LI C([ C[[[T, π, x)) is
projected onto a 2-dimensional plane using Principal Component Analysis (PCA). Fig ((a) visualizes
how Go(Z) gradually fits G(A) pre-training stage. The set of exact solutions G(A) forms
1 1-dimensional manifold (i.e. the red solid curve), and the marked points {G(m:) hesu, .19 represent
the corresponding ODE parameters used for pre-training. Each dotted curve represents a solution
set Go (m))
(Z) obtained by the neural network at the m-th iteration with the points Goem) (2) (2("))
Ugin) (+, (()) also marked on the curve. As the number of iterations m increases, the network weight = ((m) updates, making the dotted curves evolve and finally fit the red solid curve, i.e., the target manifold G(A). Biglllllllstratts illustrates the fine-tuning process for given new ODE parameter Thew A. As in Fig 3(a) the red solid curve represents the set of exact solutions G(AA), while the cyan dotted curve represents the solution Go- (Z) Gacanoy (Z) obtained by the pre-trained network. As
2 = znew updates (i.e., through fine-tuning 2), the marked point Gø- (m²) moves on the cyan dotted curve, and finally converges the approximate solution Gø- (= Go- (2(22) G((nnew).
2.3 Manifold Learning Interpretation of MAD-LM
The MAD-L method is designed for Scenario However, many parametric PDEs encountered in real applications do not fall into this scenario, especially when the parameter set A of PDEs is an andinttediimensionll function space. Simply applying MAD-L method these PDE solving problems would likely lead to unsatisfactory results. However, MAD-LM works in more general scenario, and thus has the potential of getting improved performance for a wide range of parametric PDE problems. This alternative scenario is given as follows.
Scenario 2
The solution set G(A) C U can be approximated by a set with low-dimensional structure, in the sense that there is a fmite-dimensiona
space Z R². (with 1 dim U) and Lipschitz continuous mapping G Z + U, such that ((A) is contained in the c-neighborhood of G(Z) c U, where e is a relatively small constant. In other words, for any 1 E A, there exists some Z satisfying [|G(=) G(r) c.
Appendix gives an example of G(A) that falls into Scenario but not Scenario In this new scenario, similar derivation leads to the same pre-training stage, which is used to find the initial decoder mapping Gø- Rs G. However, the fine-tuning stage, simply fine-tuning the latent vector won't t give satisfactory solution in general due the existence of the c-gap. Therefore, have
7

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New.dorat Tranle-Loura
New Sertoh Draglisdeanky
From wrw-Stratch Dengli-Searnn
15.
0/6
(a) Burgers' equation
(b) Maxwell's equations
(c) Laplace's equation
Figure 4: The mean L2 error convergence with respect to the number of training iterations.
to fine-tune the model weight with the latent vector z simultaneously, and solve the optimization problem 6). It produces new decoder Ge (z)ww) specific to the parameter Thew. An intuitive illustration is given in Fig 2bb)
Numerical Experiments
To evaluate the effectiveness of the MAD method, apply it to solve three parametric PDEs: (I) Burgers' equation with variable initial conditions; (2) Maxwell's equations with variable equation coefficients; and (3) Laplace's 's equation with variable solution domains and boundary conditions (heterogeneous PDE parameters). Accuracy of the model is measured by average relative L2 error (abbreviated L2 error) between predicted solutions and reference solutions, and we provide the mean value and the 95% confidence interval of L2 error. We compare MAD with other methods including learning from scratch (abbreviated as From-Scratch), Transfer-Learning (7), MAML [7), Reptile (18) and Pl-DeepONer [1].. For each experiment, the PDE parameters are divided into two sets: S₁ and S2- Parameters in S₁ correspond to sample tasks for pre-training, and parameters in S2 correspond to new tasks for fine-tuning. See Appendix B for the default experimental setup, and more detailed experimental setups and results for Burgers' equation, Maxwell's 's equation and Laplace's equation are given in Appendix C, D and E respectively. Unless otherwise specified, all the experiments are conducted under the MindSpore
3.1 Burgers' Equation
We consider the 1-D Burgers' equation:
du du J2u dx2' x (0, (0,
+ Ōr
u(x, = uo(x), x E (0, 1),
(13)
Eq 13) can model mieddimnsiooal flow of viscous fluid, where u is the velocity, // is the viscosity
coefficient and initial condition ux(x)) is the changing parameter of the PDE, i.c. r = un()).
The initial condition uz(x) is generated using Gaussian random field (GRF)
24 according to
un(x) ~ (0; 100 + 91) -3) with periodic boundary conditions.
Fig (() shows the mean La error of all methods as the number of training iterations increases. All methods converge to nearly the same accuracy (the mean L2 error close to 0.013) except for MAD-L. which we guess probably due to the c-gap introduced in See 2.3 In terms of convergence speed, need about 1200 iterations to converge, whereas MAML, Reptile and MAD-LM need about 200 iterations to converge. MAD-LM has the fastest convergence speed, requiring only 17% the training iterations of From-Scratch. this experiment, Transfer-Larnin does not show any advantage over From-Scrated which means that Transfer-Learning fails to obtain any useful knowledge in pre- pre-training stage.
From-Scratch and Transfer-Learnin
Pl-DeepONet can directly inference for unseen PDE parameters in S2, so it has no fine-tuning process. Table T shows the comparison of the mean La error of PI-DeepONer and MAD under different numbers of training samples in Sj. The results show that PI-DeepONer has a strong dependence on the number of training samples, and its mean L2 error remarkably high when S is small. Moreover, its mean L2 error is significantly higher than that of MAD-L or MAD-LM in all cases.
"htps//www.middsspr...ll
8

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Table 1: The mean Le error of PI- Pl-DeepONet and MAD under different numbers of samples in St.
|81]
Pl-DeepONet
MAD-L
MAD-LM
10
0.715
0.365
0.015
0.247
0.046
0.013
100
0.217
0.028
0.013
200
0.169
0.020
0.013
300
0.181
0.018
0.013
400
0.183
0.019
0.013
Frecy Sendal
Stre Strdii
Iarngfe Leaming
Bytti
Begtel MAA2-L MARDAL
MADEE
MAD
00k
0.3k
3.0%
235
50k
6Ck
5V6
100k
(a) Burgers' equation
(b) Maxwell's equations
Figure 5: The mean Le error convergence with respect to the number of training iterations for extrapolation experiments.
In the above experiments, 1/8 in S₁ and S2 come from the same GRF, tasks in the pre-training stage come from the same task distribution as the tasks in the fine-tuing stage. We investigate the extrapolation capability of MAD, that is, tasks in the fine-tuing stage come from different task distribution than those in the pre-training stage. Specifically, S is still the same above, but S2 is generated from (O; 100(-A- 25/) 2.5). 5(a) shows the results of extrapolation experiments. Since the distribution of tasks has changed, the manifold learned in the pre-training stage fits G(A) worse, so MAD-L exhibits worse accuracy than that in Fig ((a However, as with Fig ((a) the convergence speed of MAD-LM in S(a) is also better than other methods. This shows that the extrapolation capability of MAD also better than other methods in this example.
assume that the
3.2 Time-Domain Maxwell's Equations
We consider the time-domain 2-D Maxwell's equations with a point source in the transverse Electric (TE) mode (25
OE 0H, EE
OH, OH,
EE,
OE,
5)
(14)
EgEr Oy
Øt
EDE,
Polt,
Or
Dy
where Eg, E, and H. are the electromagnetic fields, J is the point source term. The equation coefficients and di the permittivity and permeability in vacuum, respectively. The equation coefficients & and H4, are the relative permittivity and relative permeability of the media, respectively. (26) uses modified PINNs method to solve Eq. (14) with fixed c, = 1 and μ4, 1. However, in this paper, (er, (4r) are variable parameters of the PDEs, i.e., "J (er- (4)), which corresponds to the media properties in the simulation region.
Fig showswws shows that all methods converge to similar accuracy (mean L2 error is close to 0.04), and MAD-LM achieves the lowest mean Le error (0.028). In terms of convergence speed, MAD-L and MAD-LM are obviously superior to other methods. It is worth noting that MAML fails to converge in pre-training stage, therefore its data is missing Fig 4(b] We guess the reason is that singularity brought by point source and computation second-order derivatives pose great difficulties in solving optimization problem. Reptile also does not show good generalization ability probably due to the same singularity problem.
We do an extrapolation experiment, where (er. Mr) in S1 comes from 51 but in the fine-tuning stage, we only consider the (er; (4)) = (7, 7) case. Because the extrapolated task does not lie in the task distribution in the pre-training stage, the point G'(now) corresponding to the extrapolated task in

---

the function space 2/ is not on the learned manifold Gg- (2), which causes MAD-L to converge to poor accuracy. Fig Findictss that MAD-LM is significantly faster than From-Scratch and Reptile also exhibits
in convergence speed while maintaining high accuracy. Notably, Transfer-L faster convergence than From-Scratch and Reptile. This because (cr: f1r) = (4,5) is randomly selected in the pre-training stage and is very close to (Gr, = (7,7) 7) in Euclidean distance.
3.3 Laplace's Equation
We consider the 2-D Laplace's equation as follows:
pu 8²u = 0, (z) EQ,
2r + 0y²
u(x,8) = g(x, y), (x, y) E as,
(15)
where the shape of Ω and boundary condition g(x, y) are the variable parameters of the PDE, i.e. ? = g0, y)). In this experiment, we use triangular domain 52 and vary the shape of Ω by randomly choosing three points the circumference of a unit circle to form the triangle. Given that h is the boundary condition on the unit circle, we use GRF to generate h ~ (0, (000,0033 (-A=((0++00)))) with periodic boundary conditions. The analytical solution of the Laplace's 's equation on the unit circle can be obtained by Fourier method. Then, we use the analytical solution on the three sides of the triangle as the boundary condition g(r,, y). The variable PDE parameters include the shape of the solution domain (the shape of the triangle) and the boundary conditions on the three sides of the triangle, so the PDE parameters here are heterogeneous. MAD can implicitly encode such heterogeneous PDE parameters as latent vectors conveniently, whereas Pl-DeepONet is unable to handle this case without further adaptations.
Fig 4(c) shows that all methods finally converge to similar accuracy (mean L2 error close to 0.001), and MAD-LM achieves the lowest mean L2 error. MAD-L, MAD-LM and Reptile show good generalization capability and excellent convergence speed, whereas Transfer-Learning and MAML do not show any advantage over From-Scratch.
Summary of experimental results. Achieving fast adaptation is the major focus of this paper, and solutions within reasonable precision need to be found. Indeed, in many control and inverse problems, a higher precision in solving the forward problem (such as parametric PDEs) does not always lead to better results. For example, solution with about 5% relative error is already enough for Maxwell's equations in certain engineering scenarios. We are therefore interested in reducing the cost of solving the PDE with a new set of parameters by using only relatively small number of iterations, in order to obtain an accurate enough solution in practice. The advantages of MAD) directly validated in the numerical experiments, convergence in the early stage of the training process. Some other applications may focus more on the final precision, and is not as sensitive to the training cost. In this alternate criterion, the superiority of the MAD method becomes less obvious in our test cases except Maxwell's 's equations, but its performance still comparable to other methods.
(especially MAD-LM)
it achieves very fast
4 Conclusions
In this paper, a novel mesh-free and unsupervised deep learning method MAD proposed for solving parametric PDEs based on meta-learning idea. A good initial model is obtained in pre-training stage to learn useful information from a set of sampled tasks, which is then used to help solve the parametric PDEs quickly in fine-tuning stage. Moreover, MAD can implicitly encode heterogeneous PDE parameters as latent vectors. The effectiveness of MAD method is analyzed from the perspective of manifold learning and verified by extensive numerical experiments.
Acknowledgments
This work supported by National Key R&D Program of China under Grant No. 2021
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Checklist
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Yes
(b) Did you describe the limitations of your work? Yes
(c) Did you discuss any potential negative societal impacts of your work? N/A
(d) Have you read the ethies review guidelines and ensured that your paper conforms them? Yes
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? N/A
(b) Did you include complete proofs of all theoretical results? N/A
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? No
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(b) Did you mention the license of the assets? N/A
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13


Reviews:
[{"review_id": "_Q1KoxJPPiZ", "reviewer": "Reviewer_Jrmb", "summary": "The paper proposes a method for learning the solutions of parameterized PDEs, where the PDE takes a set of parameters as an input. The paper introduces a neural network architecture (MAD) which takes coordinates as well as a hidden representation, which is implicitly encoded for each PDE parameters. After training, the proposed method solves optimization problem either to 1) find the best hidden representation or 2) find the best hidden representation and the best model parameters (i.e., weights and biases simultaneously).", "questions": "- how does the size of the latent vector affect the performance? how many samples (S1) would be required to learn a meaningful latent space?\n- what is the computational overhead to solve the optimization problems in MAD-L and MAD-LM, respectively? (compared to other baselines)\n- for the extrapolation task, how would the model performs if the testing parameters are generated in a following way: what if the training parameters are generated in a bounding box and the test parameters are chosen outside of the box. ", "limitations": "the proposed method, in general, is technically sound, but would be more appreciated if they provide more in-depth analysis on the experimental results, e.g., providing intuitions for setting up the size of the latent dimension, the size of the decoder, the choice of optimizers, the choice of parameter set-ups, and so on. I believe all these choices would be PDE-specific and it would be great if the authors could provide some general guidelines for making choices on those matters. Also, it would be great to see whether this method can be generalizable to more complex PDE problems. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "[+] a simple, but effective idea to extend learning PDE solutions to parameterized PDE settings\n\n[-] novelty: learning hidden representations of parameterized PDEs in training time and exploring the latent space for finding the best hidden representation for the new parameter instances have been studied in reduced-order modeling context (Eq (7.3) in Lee and Carlberg, JCP, 2020), which can be considered as an equivalent idea of MAD-L. Although the context and the optimization objective (Eq.(5)) are different, the idea of exploring latent space for the best hidden representation of parameterized PDEs has been out there. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "KD1qaevGdI_", "reviewer": "Reviewer_znr2", "summary": "This paper introduces a manifold learning approach for speeding up solutions to parametric PDEs, inspired by DeepSDF [22]. They associate a latent vector for each each instance of the PDE, and learn a solution manifold using an initial dataset of PDE instances. During inference, they either search for the closest latent vector (using gradient-based optimization) (MAD-L) or fine-tune both the network parameters alongside the latent vector (MAD-LM).\n\nThey report strong performance improvements compared to Meta-Learning, PINNs from scratch, and DeepONets.", "questions": "For the experiments with MAD, you had to introduce extra parameters in the NN to take the latent vector as input. Does this significantly increase parameter count? Could you give the number of NN parameters for \"From-Scratch\" experiments vs \"MAD-L/LM\" experiments?\n\nIt would be really nice to see a comparison of these methods to more standard numerical methods (such as FEM/spectral methods). it would be interesting to see if MAD-L can get close to or even surpass more classical methods in terms of performance.\n\nQuality of life nits: Could you add descriptions of the figures into the captions themselves as well? It becomes difficult to scroll back and forth.", "limitations": "It was really nice seeing the cases where MAD-L failed. I think comparisons with more standard numerical methods could also make it clear where this method stands in terms of potential limitations compared to state of the art.\n\nPotential negative societal impact was not assessed.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper attacks an important problem. Often times we have to solve many versions of a PDE, either for design optimization or parameter discovery. One of the strengths of using neural networks as an ansatz for PDE solutions is that we can use techniques such as meta-learning to speed up solution process. This paper adapts methods from the shape representation literature to this problem, and shows consistent performance improvements.\n\nThe contribution is simple but works well, so I think it is solid. I like the sections on the manifold learning intuition, especially the simple ODE example, as I think it really motivates why the algorithm works.\n\nIt would be nice to see results from larger-scale experiments. Could you demonstrate a case-study where you would want to solve a sequence of PDEs, and your meta-learning setup gives large performance gains? I think this would help motivate the problem of parametric PDEs much better.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "7XiIq-zmH_7", "reviewer": "Reviewer_1XCo", "summary": "This paper propose Meta-Auto-Decoder (MAD), a mesh-free and unsupervised deep learning method that uses pretraining by encoding PDE parameters to latent vectors. MAD has a manifold learning interpretation. Experiments show that MAD is faster than other deep learning-based methods.", "questions": "1. When pretraining, what data is used? In paper, it says 'randomly-generated'? Does it mean both target and pretrain data are randomly-generated? What knowledge is shared among these data?\n2. When comparing effienciency, is the pretraining stage been taken into consideration for MAD?", "limitations": "So far no. ", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. Well-motivated and clear presentation of contributions\n2. Well-validated experimental result\n\nWeaknesses:\n1. Many concepts are shared with Meta-Mgnet and moreover not empirically compared with Meta-MgNet\n2.  Pretraining part is a bit unclear to me (detailed in the questions part)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept