Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains
Proposes a transferable deep learning framework combining GFNet and Mosaic predictor for unseen PDE domains, achieving 3 orders speedup.
Key Findings
Methodology
The framework comprises GFNet, a neural network trained on small domains to learn PDE solutions under arbitrary boundary conditions, and the Mosaic flow (MF) predictor, which iteratively assembles GFNet inferences across large, unseen domains using the continuous Schwarz algorithm. GFNet employs architecture designs like linear-preserving fully connected layers to enhance accuracy, while MF leverages domain decomposition and overlapping subdomains to maintain spatial continuity. Once trained, GFNet is reused for various boundary conditions and domain sizes, eliminating retraining. Experiments on Laplace and Navier-Stokes equations show GFNet reduces errors by 50-70% compared to traditional networks, and MF scales solutions up to 1200Γ and 12Γ larger domains with errors reduced by nearly 6 times, outperforming DeepONet and FNO.
Key Results
- GFNet achieves twice the accuracy of conventional fully connected networks in solving Laplace equations on unseen shapes, with error reductions exceeding 50%. In Navier-Stokes, boundary condition variations lead to 70% error decrease. MF predictor extends solutions to domains 1200Γ (Laplace) and 12Γ (NS) larger, maintaining high precision. Once trained, the model predicts flow features in new geometries rapidly, with error metrics surpassing existing methods by nearly sixfold. This approach enables efficient, scalable PDE solutions without retraining, drastically reducing computational costs.
- The framework demonstrates strong generalization across boundary conditions and geometries, addressing the domain dependency issues of PINNs. It combines physical insights with deep learning, enabling high-fidelity, transferable PDE approximations. Ablation studies confirm architecture choices and domain decomposition strategies are critical for performance.
- Experimental results validate that GFNet and MF outperform baseline models in accuracy and scalability, with errors consistently below benchmarks across diverse test cases. The approach is robust for linear and nonlinear PDEs, promising broad applicability in scientific computing.
Significance
This work addresses the longstanding challenge of transferability in deep learning-based PDE solvers, enabling once-trained models to be applied across diverse geometries and boundary conditions. It significantly reduces computational costs, facilitating large-scale simulations in engineering and physical sciences. By integrating physical principles into neural architectures and domain decomposition algorithms, the framework enhances both accuracy and efficiency, paving the way for real-time, high-fidelity simulations in complex systems such as aerodynamics, climate modeling, and geophysics. Its ability to generalize without retraining marks a major step toward practical, scalable scientific machine learning solutions.
Technical Contribution
The paper introduces GFNet, a physics-informed neural network architecture tailored to preserve PDE linearity and physical constraints, along with the MF predictor, which employs domain decomposition and the continuous Schwarz algorithm for scalable, high-accuracy solutions. This combination allows a single training phase to serve multiple domain sizes and boundary conditions, overcoming the domain-specific limitations of existing PINNs and operator learning models. The framework's theoretical foundation ensures convergence and accuracy guarantees, while empirical results demonstrate substantial improvements over state-of-the-art methods like DeepONet and FNO, with errors reduced by nearly six times and speedups of three orders of magnitude.
Novelty
This is the first work to integrate PDE-structure-aware architecture design with a domain decomposition-based iterative assembly (via Schwarz method) in a deep learning context, enabling transferability across arbitrary domain sizes and boundary conditions. Unlike prior models limited to fixed geometries, GFNet combined with MF predictor achieves scalable, accurate solutions in unseen domains, breaking the domain-size and shape restrictions that have hampered deep learning PDE solvers. The approach innovatively merges classical domain decomposition algorithms with modern neural networks, establishing a new paradigm for scalable, transferable scientific ML.
Limitations
- The method's performance may degrade in highly nonlinear, multi-physics, or discontinuous boundary conditions, where physical constraints are harder to encode or learn. Extreme geometries or irregular domain decompositions could challenge the current domain partitioning strategy. Additionally, reliance on high-fidelity simulation data for training boundary conditions limits applicability in some real-world scenarios. Further research is needed to enhance robustness, automate domain partitioning, and incorporate adaptive boundary condition generation.
Future Work
Future directions include extending the framework to multi-physics and multi-scale problems, improving adaptive domain decomposition strategies, and integrating uncertainty quantification. Developing physics-informed boundary condition generators and exploring graph neural network architectures could further enhance generalization. Additionally, applying the approach to real-world large-scale engineering problems, such as climate modeling or aerospace design, will test its robustness and practical utility. The ultimate goal is to create a universally transferable, real-time PDE solver capable of handling complex, dynamic systems with minimal human intervention.
AI Executive Summary
Partial differential equations (PDEs) are fundamental in modeling physical phenomena across science and engineering, yet traditional numerical methods like finite element and finite difference are computationally intensive, especially for large or complex domains. Recent advances in deep learning, including physics-informed neural networks (PINNs) and operator learning models like DeepONet and FNO, have shown promise in approximating PDE solutions. However, these models typically suffer from poor transferability: they need to be retrained whenever the domain shape, size, or boundary conditions change, limiting their practical use in large-scale engineering problems.
This paper introduces a novel, scalable framework that overcomes these limitations by combining a specially designed neural network architecture, GFNet, with an iterative assembly algorithm, the Mosaic flow (MF) predictor. GFNet is trained once on small, simple domains to learn the solution of PDEs under arbitrary boundary conditions, leveraging physical insights such as linearity preservation. The MF predictor employs domain decomposition and the continuous Schwarz algorithm to assemble multiple GFNet inferences into a coherent solution over large, complex domains, without retraining. This approach enables the model to generalize across domain sizes, shapes, and boundary conditions, a feat previously unattainable.
Experimental results demonstrate that GFNet achieves twice the accuracy of conventional fully connected networks in solving Laplace equations on unseen geometries, with errors reduced by over 50%. For Navier-Stokes equations, the method maintains high accuracy even when the domain is scaled up 1200 times or the boundary conditions vary significantly. The combined framework reduces the error by nearly sixfold compared to DeepONet and FNO, while delivering speedups of three orders of magnitude. Once trained, the model can predict solutions in new scenarios instantly, greatly accelerating large-scale simulations.
This work significantly advances the field of scientific machine learning by providing a practical, transferable PDE solver that can be applied across diverse applications, including aerodynamics, climate modeling, and geophysics. It bridges the gap between high-fidelity numerical methods and scalable deep learning, offering a new paradigm for real-time, accurate, and generalizable PDE solutions. Despite some limitations in handling highly nonlinear or irregular geometries, the framework opens numerous avenues for future research, such as multi-physics integration, adaptive domain partitioning, and uncertainty quantification, promising a transformative impact on computational science and engineering.
Deep Dive
Abstract
Physics-informed neural networks (PINNs) are increasingly employed to replace/augment traditional numerical methods in solving partial differential equations (PDEs). While state-of-the-art PINNs have many attractive features, they approximate a specific realization of a PDE system and hence are problem-specific. That is, the model needs to be re-trained each time the boundary conditions (BCs) and domain shape/size change. This limitation prohibits the application of PINNs to realistic or large-scale engineering problems especially since the costs and efforts associated with their training are considerable. We introduce a transferable framework for solving boundary value problems (BVPs) via deep neural networks which can be trained once and used forever for various unseen domains and BCs. We first introduce genomic flow network(GFNet), a neural network that can infer the solution of a BVP across arbitrary BCson a small square domain called genome. Then, we proposed mosaic flow(MF) predictor, a novel iterative algorithm that assembles the GFNet's inferences for BVPs on large domains with unseen sizes/shapes and BCs while preserving the spatial regularity of the solution. We demonstrate that our framework can estimate the solution of Laplace and Navier-Stokes equations in domains of unseen shapes and BCs that are, respectively, 1200 and 12 times larger than the training domains. Since our framework eliminates the need to re-train models for unseen domains and BCs, it demonstrates up to 3 orders-of-magnitude speedups compared to the state-of-the-art.