[Paper Review] One-shot learning for solution operators of partial differential equations

Jiao, Anran, Haiyang He+3 more|arXiv (Cornell University)|Apr 6, 2021

Model Reduction and Neural Networks13 references4 citations

TL;DR

This paper proposes the first one-shot learning method for PDE solution operators, leveraging local solution operators trained on a single PDE solution by decomposing the domain into small subdomains. Using mesh-based fixed-point iteration or meshfree local-solution-operator informed neural networks (LOINNs), the method achieves strong generalization across linear and nonlinear PDEs with minimal data, demonstrating low error rates even under noisy conditions.

ABSTRACT

Learning and solving governing equations of a physical system, represented by partial differential equations (PDEs), from data is a central challenge in a variety of areas of science and engineering. Traditional numerical methods for solving PDEs can be computationally expensive for complex systems and require the complete PDEs of the physical system. On the other hand, current data-driven machine learning methods require a large amount of data to learn a surrogate model of the PDE solution operator, which could be impractical. Here, we propose the first solution operator learning method that only requires one PDE solution, i.e., one-shot learning. By leveraging the principle of locality of PDEs, we consider small local domains instead of the entire computational domain and define a local solution operator. The local solution operator is then trained using a neural network, and utilized to predict the solution of a new input function via mesh-based fixed-point iteration (FPI), meshfree local-solution-operator informed neural network (LOINN) or local-solution-operator informed neural network with correction (cLOINN). We test our method on diverse PDEs, including linear or nonlinear PDEs, PDEs defined on complex geometries, and PDE systems, demonstrating the effectiveness and generalization capabilities of our method across these varied scenarios.

Motivation & Objective

• To address the challenge of learning PDE solution operators with minimal data, particularly when only one solution is available.
• To overcome the data inefficiency of existing deep learning methods for PDEs, which typically require large datasets.
• To develop a method that generalizes well across different initial and forcing conditions using only one reference solution.
• To leverage the locality of PDEs to enable scalable, stable, and accurate solution prediction in new conditions.

Proposed method

• Decompose the computational domain into small local subdomains to learn localized solution operators.
• Train a neural network to map local boundary values and forcing terms to the solution at a central point within each subdomain.
• Use mesh-based fixed-point iteration (FPI) to iteratively enforce global boundary and initial conditions while satisfying local PDE constraints.
• Introduce two meshfree variants—LOINN and cLOINN—using randomly sampled points to improve stability and robustness.
• Formulate the solution operator learning as a one-shot problem by selecting a reference forcing function and its corresponding solution.
• Apply the method to both linear and nonlinear PDEs, including the diffusion equation and nonlinear diffusion-reaction equations, on structured and unstructured grids.

Experimental results

Research questions

• RQ1 Can a solution operator for a PDE be accurately learned from only one solution, without prior knowledge of the PDE form?
• RQ2 How does the choice of local domain size and structure affect the accuracy and convergence of the solution prediction?
• RQ3 Can meshfree LOINN and cLOINN methods outperform mesh-based fixed-point iteration in stability and robustness under noisy conditions?
• RQ4 How well does the method generalize to new forcing functions near the reference solution, especially with increasing noise levels?
• RQ5 What is the impact of grid resolution on the performance of one-shot local solution operator learning?

Key findings

• For the linear diffusion equation with a 4-node local domain, FPI and cLOINN achieved relative $L^2$ errors below 1.5% when noise level $\sigma = 0.10$, outperforming LOINN.
• With a 6-node local domain, FPI diverged, but cLOINN maintained stable performance with errors below 3.5% at $\sigma = 0.50$, demonstrating superior robustness.
• For the nonlinear diffusion-reaction equation, the method achieved $L^2$ relative errors around 10% even on coarse grids (21×21), indicating strong generalization.
• Error increased with higher noise levels ($\sigma$), but all methods remained stable and predictive, with cLOINN showing the most consistent performance.
• The method generalized effectively to new forcing functions $f = 0.9\sin(2\pi x) + \Delta f$ across multiple test cases, confirming its applicability to real-world scenarios with perturbations.
• LOINN and cLOINN showed improved stability over FPI in nonlinear and high-noise regimes, validating the advantage of meshfree, neural network-based coupling.