One-shot learning for solution operators of partial
differential equations

Anran Jiao
Department of Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104
Haiyang He, Rishikesh Ranade, Jay Pathak
Ansys Inc.
San Jose, CA 95134
Lu Lu
Department of Chemical and Biomolecular Engineering
University of Pennsylvania
Philadelphia, PA 19104
lulu1@seas.upenn.edu

Abstract
Discovering governing equations of a physical system, represented by partial
differential equations (PDEs), from data is a central challenge in a variety of areasof science and engineering. Current methods require either some prior knowledge(e.g., candidate PDE terms) to discover the PDE form, or a large dataset to learn asurrogate model of the PDE solution operator. Here, we propose the first solution
operator learning method that only needs one PDE solution, i.e., one-shot learning.
We first decompose the entire computational domain into small domains, where
we learn a local solution operator, and then we find the coupled solution via either
mesh-based fixed-point iteration or meshfree local-solution-operator informed
neural networks. We demonstrate the effectiveness of our method on different
PDEs, and our method exhibits a strong generalization property.

1Introduction

Discovering governing equations of a physical system from data is a central challenge in a varietyof areas of science and engineering. These governing equations are usually represented by partial
differential equations (PDEs). For the first scenario, where we know all the terms of the PDE andonly need to infer unknown coefficients from data, many effective methods have been proposed. For
example, we can enforce physics-based constraints to train neural networks that learn the underlying
physics [1, 2, 3, 4, 5, 6, 7, 8]. In the second scenario, we do not know all the PDE terms, but wehave a prior knowledge of all possible candidate terms, several approaches have also developed
recently [9, 10, 11].
In a general setup, discovering PDEs only from data without any prior knowledge is much moredifficult. To address this challenge, instead of discovering the PDE in an explicit form, in mostpractical cases, it is sufficient to have a surrogate model of the PDE solution operator that can predict
PDE solutions repeatedly for different conditions (e.g., initial conditions). Very recently, several
approaches have been developed to learn PDE solution operators by using neural networks such as
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

DeepONet [12, 13, 14] and Fourier neural operator [15, 13]. However, these approaches require large
amounts of data to train the networks.
In this work, we propose a novel approach to learn PDE solution operators from only one data
point, i.e., one-shot learning. To our knowledge, the use of one-shot learning in this space is very
limited. [16] and [17] used few shot learning to learn PDEs and applied it to face recognition. Our
method first leverages the locality of PDEs and uses neural networks to learn the system governed by
PDEs at a small computational domain. Then for a new PDE condition, we couple all local domains
to find the PDE solution. A mesh-based fixed-point iteration (FPI) approach is proposed to learn
the PDE solution that satisfies the boundary/initial conditions and local PDE constraints. We also
propose two versions of local-solution-operator informed neural networks (LOINNs), which aremeshfree, to improve the stability and robustness of finding the solution. Moreover, our one-shot
local learning method can extend to multi-dimensional, linear and non-linear PDEs. In this paper, we
describe, in detail, the one-shot local learning method and demonstrate on different PDEs for a rangeof conditions.

2

Methods

We first introduce the problem setup of learning solution operators of PDEs and then present our
one-shot learning method.
2.1

Learning solution operators of PDEs

We consider a physical system governed by a PDE defined on a spatio-temporal domain Ω ⊂ Rd :F[u(x); f (x)] = 0,

x = (x1 , x2 , . . . , xd ) ∈ Ωwith suitable initial and boundary conditions B(u(x), x) = 0. u(x) is the solution of the PDE andf (x) is a forcing term. The solution u depends on f , and thus we define the solution operator as
G : f (x) 7→ u(x). For nonlinear PDEs, G is a nonlinear operator.
In many problems, the PDE of a physical system is unknown or computationally expensive to solve,
and instead, sparse data representing the physical system is available. Specifically, we consider a|T |dataset T = {(fi , ui )}i=1 , and (fi , ui ) is the i-th data point, where ui = G(fi ) is the PDE solution
for fi . Our goal is to learn G from the training dataset T , such that for a new f , we can predict thecorresponding solution u = G(f ). When T is sufficiently large, then we can learn G straightforwardly
by using neural networks, whose input and output are f and u, respectively. Many networks havebeen proposed in this manner such as DeepONet [12] and Fourier neural operator [15]. In thisstudy, we consider an extreme scenario where we have only one data point for training, i.e., one-shotlearning with |T | = 1, and we let T = {(fT , uT )}. Learning from only one data point is impossible
in general, and here we consider that T is not given, and we can select fT . In addition, instead oflearning G for the entire input space, we only predict f in a neighborhood of some f0 , where weknow the solution u0 = G(f0 ).
2.2

One-shot learning based on locality

To overcome the difficulty of training a machine learning model based on only one data point, wefirst consider the fact that derivatives and PDEs are defined locally, i.e., the same PDE is satisfied in
an arbitrary small domain inside Ω.
To demonstrate the idea, we consider a mesh node at the location x∗ (the red node in Fig. 1). In order topredict the solution u(x∗ ), instead of considering the entire computational domain, we only consider asmall domain Ω̃ surrounding x∗ . We term the points inside Ω̃ as auxiliary points of x∗ . If we know thesolution u at the boundary of Ω̃ (∂ Ω̃) and f within Ω̃, then u(x∗ ) is determined by the PDE. Here, we
use a neural network to represent this relationship G̃ : {u(x) : x ∈ ∂ Ω̃} ∪ {f (x) : x ∈ Ω̃} 7→ u(x∗ ).
In addition, considering the flexibility of neural networks, we may use other local information
as network inputs. For example, we can only use the value of f at x∗ and the solutions of all
auxiliary points inside Ω̃ as network inputs: G̃ : {u(x) : x ∈ Ω̃ and x ̸= x∗ } ∪ {f (x∗ )} 7→ u(x∗ ).
2

The size and shape of Ω̃ are also hyperparameters to be chosen. Because
the network learns for a small local domain, by traversing the entire domainΩ, we can generate many input-output pairs for training the network, whichmakes it possible to learn a network from only one PDE solution. We willcompare the performance of several different choices of network inputs
in our numerical experiments.
We first train a neural network G̃ with the dataset T = {(fT , uT )}. Using
the pre-trained model G̃, in the second stage of our method, we propose
three approaches to predict the solution of a new f = f0 + ∆f . Since Figure 1: PDE in a lothe inputs of the pre-trained neural network also include the solution to be cal domain Ω̃. A neuralpredicted, we cannot predict solution for f directly.
network G̃ is trained tolearn the mapping from
Prediction via a fixed-point iteration (FPI). We first propose a mesh- all or some of u(x) andbased iterative approach (Algorithm 1). Because f is close to f0 ,we use f (x) for x ∈ Ω̃ to
u0 as the initial guess of u, and then in each iteration, we apply the trained u(x∗ ).
network on the current solution as the input to get a new solution. Whenthe solution is converged, u and f are consistent with respect to the local
operator G, and thus the current u is the solution of our PDE.
Algorithm 1: Predicting the solution u = G(f ) for a new f via FPI.
Initiate: u(x) ← u0 (x) for all x ∈ Ω;
2 while u has not converged do3
for x ∈ Ω do4û(x) ← G̃(the inputs of u and f in Ω̃(x));
5
Apply boundary and initial conditions to û(x);6Update: u(x) ← û(x) for all x ∈ Ω;
1

Prediction via a local-solution-operator informed neural network (LOINN). We also propose aneural-network based approach, which is meshfree and has more flexibility to handle boundary/initialconditions and training points inside the computationial domain. We construct a neural networkthat takes the coordinates x as the input, and output the approximated solution û(x). To train the
network, we constraint û(x) on G̃ via the loss function defined as the discrepancy between û(x) andG̃(û and f in Ω̃(x)):
2
1 X
1 X
L=
û(x) − G̃(û and f in Ω̃(x)) +∥B(û(x), x)∥22 .
|Tl |
|Tb |
x∈Tl

x∈Tb

Tl and Tb are the data points in the domain constrained by the local solution operator G̃ and on theboundary, respectively. The network architecture is shown in Fig. 4.
Prediction via a local-solution-operator informed neural network with correction (cLOINN).
To improve the performance of LOINN, we choose the approximated solution as û(x) = N (x) +u0 (x), where N (x) is the neural network output. We show the architecture of cLOINN in Fig. 5.

3

Results

In this section, we show the effectiveness of our proposed method for a few problems, and compare
the accuracy of different choices of the local solution operator G̃ and training data. To generate thedataset T , we use fT (x) = frandom (x) + f0 (x), where frandom (x) ∼ GP(0, k(x1 , x2 )) is randomly
sampled from a mean-zero Gaussian random field (GRF) with the covariance kernel k(x1 , x2 ) =σ 2 exp(−∥x1 − x2 ∥2 /2l2 ) (σ: standard deviation; l: correlation length). The numerical solution is
obtained via the finite difference method in an equispaced dense grid with a grid size hd . However,
we learn G in a coarser grid with a step size hc . The choices of σ, l, hd , and hc for each example arelisted in Table 2. After the pre-trained model G̃ is obtained, we use FPI, LOINN, and cLOINN to3

predict a new f = f0 + ∆f , in which ∆f is sampled from a GRF with a correlation length l = 0.1
and different standard deviation σ. In this study, we only consider the structured equispaced grid todemonstrate our method, but LOINN and cLOINN allow us to randomly sample points in the domain
and on the boundaries.
We first demonstrate the capability of our method by a pedagogical example of a one-dimensional
Poisson equation in Appendix C.
3.1

Linear diffusion equation
2∂ u
The second equation we consider is a linear diffusion equation ∂u∂t = D ∂x2 + f (x) with zeroboundary and initial conditions, where D = 0.01 is the diffusion coefficient, and the solution
operator is G : f 7→ u. We consider two candidate local solution operators (Fig. 2): (A) the simplest
local domain with 4 nodes G̃1 : {u(xi , tj−1 ), u(xi−1 , tj ), u(xi+1 , tj ), f (xi , tj )} 7→ u(xi , tj ), and(B) a larger local domain with 6 nodes G̃2 : {u(xi−1 , tj−1 ), u(xi , tj−1 ), u(xi+1 , tj−1 ), u(xi−1 , tj ),u(xi+1 , tj ), f (xi , tj )} 7→ u(xi , tj ).

We want to predict the solution for a new f =0.9 sin(2πx) + ∆f . When we consider the local solution
operator G̃1 , for a fixed σ, FPI and cLOINN both work
well and outperform LOINN (Table 1). When it comes toG̃2 , LOINN and cLOINN give us stable results, but FPI Figure 2: Local domains Ω̃ of the linear
diverges. We also observe that the errors increase when diffusion-reaction equation. Domains
σ is larger. The details of error convergence are shown in with (A) 4 nodes and (B) 6 nodes.
Figs. 10 and 11.
Table 1: L2 relative errors of different approaches for linear diffusion equation test on 100
random f .
FPI

σ
0.10
0.30
0.50
0.80
3.2

LOINN

cLOINN

G̃1

G̃2

G̃1

G̃2

G̃1

G̃2

0.51 ± 0.14%
1.30 ± 1.08%
2.77 ± 2.38%
5.82 ± 6.31%-

4.95 ± 2.30%
5.06 ± 2.52%
6.35 ± 2.52%
9.22 ± 5.64%

0.78 ± 0.31%
1.15 ± 1.60%
3.05 ± 2.51%
5.78 ± 6.60%

1.14 ± 0.63%
1.76 ± 1.12%
3.23 ± 2.49%
6.39 ± 7.73%

1.49 ± 0.65%
3.31 ± 1.93%
5.56 ± 2.99%
8.87 ± 7.62%

Nonlinear diffusion-reaction equation2∂ u
2
We consider a nonlinear diffusion-reaction equation with a source term f (x): ∂u∂t = D ∂x2 + ku +f (x), x ∈ [0, 1], t ∈ [0, 1], with zero initial and boundary conditions, where D = 0.01 is the diffusion
coefficient, and k = 0.01 is the reaction rate. The solution operator is the mapping from f (x) tou(x, t).

We use the same G̃1 as the previous example. In this experiment, the numerical solutions are obtained
from a denser grid of 1001×1001, but we learn G̃1 on a coarser grid of 101×101, 51×51, 26×26,
21×21, or 16×16. Here f0 = 0.9 sin(2πx). We use FPI, LOINN and cLOINN, and compare L2
relative errors with different σ and resolutions. With higher resolution, all approaches perform better(Fig. 3). The errors can achieve approximately 10% even on a coarser grid of 21×21 (Tables 5, 6, 7,
8, and 9), which demonstrates the robustness and generalizability of our proposed method.

4

Conclusions

Learning solution operators of partial differential equations (PDEs) usually requires a large amountof training data. In this study, we propose, to the best of our knowledge, the first one-shot method tolearn solution operators from only one PDE solution. In future, we will carry out further validation
on different types of PDEs, extend the method to unstructured meshes, and improve our approaches
for faster convergence and better computational efficiency.
4Figure 3: L2 relative error of different test functions with σ = 0.10, 0.30, 0.50, 0.80 when using
different grid resolution for the nonlinear diffusion-reaction equation. (A) FPI. (B) LOINN. (C)cLOINN.

Broader impacts
In this paper, we propose the first one-shot method to learn solution operators of PDEs. With only
one data point, our work points out a direction to improve and accelerate scientific research related to
discovering governing equations of physics systems. In the meanwhile, when employing our method,
we also need to consider ethical implications and prevent abusing machine learning technologies inapplications of science and engineering.

Acknowledgments
This work was supported by the U.S. Department of Energy [DE-SC0022953].

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were chosen)? [Yes] Yes, all the details are provided in the main paper.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Yes, we report the standard deviation in the tables.
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Schematic of LOINN and cLOINN

Figs. 4 and 5.

Figure 4: The architecture of LOINN.

Figure 5: The architecture of cLOINN.

B

Hyperparameters

Tables 2 and 3.
Table 2: Hyperparameters for numerical experiments.

l
σ
hd
hc

C

Appendix C

Section 3.1

Section 3.2

0.01
0.5
0.1
0.01

0.01
0.1
0.1
0.01

0.01
0.1
0.1
0.01, 0.02, 0.04, 0.05, 0.07

1D Poisson equation

In this section, we consider a pedagogical example of a one-dimensional Poisson equation ∆u =f (x), x ∈ [0, 1], with the zero Dirichlet boundary condition u(0) = u(1) = 0, and the solution
operator is G : f 7→ u. We choose the simplest local operator G̃1 : {u(xi−1 ), u(xi+1 ), f (xi )} 7→
u(xi ) and G̃2 : {u(xi−1 ), u(xi+1 ), f (xi−1 ), f (xi+1 ), f (xi )} 7→ u(xi ) with 3 nodes. The training
dataset T only has one data point (fT , G(fT )), and one example of T is shown in Fig. 6.
7

Table 3: Hyperparameters of the neural networks.

Appendix C
Section 3.1
Section 3.2

G̃1
G̃2
G̃1
G̃2
G̃

Depth

Width

Optimizer#Iterations

2
2
2
2
2

64
32
32
32
64

Adam + L-BFGS
Adam + L-BFGS
Adam + L-BFGS
Adam + L-BFGS
Adam + L-BFGS

200000200000100000100000150000We assume that we have the solution u0 for f0 = sin(2πx). We show examples of f = f0 + ∆f in
Fig. 7, and the black solid line is f0 . When σ is larger, there is a greater difference between f0 and f .
We report the geometric mean of L2 relative errors of all cases in Table 4. The comparison of results
using different σ is shown in Fig. 8. When σ = 0.02, all approaches reached L2 relative error of
about 0.1%. It is observed that in this experiment the local solution operator G̃1 outperforms G̃2 from
Fig. 9. As expected, when σ is smaller, the performance is better. cLOINN outperforms LOINN, andperforms as well as FPI.
Table 4: L2 relative errors of different approaches for 1D Poisson equation test on 100 random
∆f .
FPI

σ
0.02
0.05
0.10
0.15

LOINN

cLOINN

G̃1

G̃2

G̃1

G̃2

G̃1

G̃2

0.83 ± 0.59%
1.90 ± 1.79%
3.88 ± 3.64%
6.09 ± 6.07%

1.13 ± 0.43%
1.79 ± 1.39%
3.73 ± 2.97%
6.10 ± 4.70%

0.90 ± 0.59%
1.90 ± 1.79%
3.90 ± 3.64%
6.10 ± 6.07%

1.14 ± 0.43%
1.83 ± 1.41%
3.82 ± 2.94%
6.08 ± 4.71%

0.79 ± 0.55%
1.87 ± 1.80%
3.94 ± 4.24%
6.20 ± 6.34%

1.12 ± 0.43%
1.78 ± 1.39%
3.74 ± 2.97%
6.18 ± 7.86%

Figure 6: One example of the training data for the 1D Poisson equation. (A) A random fT
generated from GRF. (B) The corresponding solution uT .

D

Linear diffusion equation

Fig. 10 and Table 5.

E

Nonlinear diffusion-reaction equation

Tables 5, 6, 7, 8 and 9.

8

Figure 7: Examples of some random f = f0 + ∆f with ∆f sampled from GRF of different σ
and l = 0.1. (A) σ = 0.02. (B) σ = 0.05. (C) σ = 0.10. (C) σ = 0.15.

Figure 8: Learning solution of the 1D Poisson equation. The L2 relative errors tested on 100
random ∆f sampled from GRF of various σ and l = 0.1 with local operator (A to C) G̃1 and (D toF) G̃2 . (A and D) FPI. (B and E) LOINN. (C and F) cLOINN.

Figure 9: Learning solution of the 1D Poisson equation with different local operators G̃. The L2
relative errors tested on 100 random ∆f sampled from GRF of σ = 0.02 and l = 0.1 using (A) FPI,
(B) LOINN, and (C) cLOINN.
Table 5: L2 relative errors of different approaches for nonlinear diffusion-reaction equation teston 100 random ∆f . We learn the local operator on a coarser mesh size of 101.
σ

FPI

LOINN

cLOINN

0.10
0.30
0.50
0.80

0.32 ± 0.08%
0.91 ± 0.51%
1.81 ± 1.37%
3.32 ± 3.54%

0.52 ± 0.20%
1.13 ± 0.60%
2.26 ± 1.46%
2.89 ± 2.16%

0.49 ± 0.21%
1.02 ± 0.62%
2.30 ± 1.52%
2.96 ± 2.39%

9

Figure 10: Learning solution of the linear diffusion equation with different local operators G̃.
The L2 relative errors tested on 100 random f sampled from GRF of σ = 0.10 and l = 0.1 using (A)LOINN and (B) cLOINN.

Figure 11: Learning solution of the linear diffusion equation. The L2 relative errors tested on 100
random ∆f sampled from GRF of various σ and l = 0.1 with local operator (A to C) G̃1 and (D andE) G̃2 . (A) FPI. (B and D) LOINN. (C and E) cLOINN.
Table 6: L2 relative errors of different approaches for nonlinear diffusion-reaction equation teston 100 random f . We learn the local operator on a coarser mesh size of 51.
σ

FPI

LOINN

cLOINN

0.10
0.30
0.50
0.80

0.96 ± 0.30%
2.50 ± 0.78%
4.21 ± 1.47%
6.34 ± 2.59%

1.02 ± 0.30%
2.57 ± 0.77%
4.28 ± 1.46%
6.45 ± 2.57%

1.02 ± 0.29%
2.54 ± 0.78%
4.34 ± 1.52%
6.63 ± 2.94%

Table 7: L2 relative errors of different approaches for nonlinear diffusion-reaction equation teston 100 random f . We learn the local operator on a coarser mesh size of 26.
σ

FPI

LOINN

cLOINN

0.10
0.30
0.50
0.80

1.23 ± 0.35%
3.58 ± 1.27%
5.88 ± 1.94%
8.15 ± 3.26%

1.19 ± 0.34%
3.60 ± 1.25%
5.36 ± 1.85%
8.12 ± 3.50%

1.16 ± 0.34%
3.54 ± 1.29%
5.59 ± 1.84%
8.52 ± 4.18%

10

Table 8: L2 relative errors of different approaches for nonlinear diffusion-reaction equation teston 100 random f . We learn the local operator on a coarser mesh size of 21.
σ

FPI

LOINN

cLOINN

0.10
0.30
0.50
0.80

2.32 ± 0.59%
6.03 ± 2.06%
10.41 ± 3.40%
13.18 ± 4.94%

2.44 ± 0.64%
6.42 ± 1.92%
10.38 ± 4.13%
10.66 ± 4.37%

2.40 ± 0.67%
6.29 ± 1.94%
10.35 ± 4.29%
10.59 ± 4.52%

Table 9: L2 relative errors of different approaches for nonlinear diffusion-reaction equation teston 100 random f . We learn the local operator on a coarser mesh size of 16.
σ

FPI

LOINN

cLOINN

0.10
0.30
0.50
0.80

8.58 ± 0.27%
10.93 ± 1.84%
13.41 ± 3.51%
17.35 ± 5.26%

8.60 ± 0.27%
10.94 ± 1.85%
13.42 ± 3.41%
17.37 ± 5.25%

8.58 ± 0.27%
10.93 ± 1.84%
13.42 ± 3.40%
17.36 ± 5.26%

11