Neural solvers enable fast adaptable solutions for differential problems
Single-condition neural solvers encode transferable response spaces for parametric differential equations
Machine Learning
Summary
Solving complex differential equations across many conditions usually needs lots of data or slow training. The authors found that training a neural network on one condition creates a useful space to represent related solutions under different conditions. They developed methods to combine these spaces from multiple trainings to quickly and accurately solve new problems. Their approach works well across several systems and is faster and more precise than previous methods.
What this means in practice
- •For computational engineers: Accelerate simulation adjustments under varying physical conditions with reusable neural response models.
- •For software developers for scientific computing: Implement adaptive neural solvers that reduce training time and improve accuracy for diverse parameter settings in PDE-based software.
Authors
Wenbo Cao, Weiwei Zhang
Abstract
Operator learning for parametric partial differential equations (PDEs) typically builds global models over prescribed domains, requiring cross-condition data or costly physics-constrained training. Here we show that the output Jacobian of a neural solution model trained at one condition defines a reusable response space for cross-condition solution variations. We introduce Linearized Subspace Transfer (LST) to exploit this space and recover target solutions by minimizing the target PDE-system residual over response-space coordinates. Because any single response space has finite coverage, Active Transfer Modeling (ATM) uses post-transfer residuals as coverage indicators to selectively acquire response spaces from additional single-condition models. Across six systems, single-condition response spaces supported cross-condition transfer, with enrichment improving accuracy when added spaces expanded representation capacity. Relative to evaluated physics-informed operator baselines, ATM reduced error and offline construction cost, with orders-of-magnitude accuracy gains in representative cases and millisecond-to-second target adaptation. These results establish neural solvers as reusable local parametric models.