aiqbench analysis
This research introduces Linearized Subspace Transfer (LST) and Active Transfer Modeling (ATM), techniques that allow neural solvers to generalize across different conditions without requiring costly cross-condition training data. By treating the output Jacobian of a single-condition model as a reusable response space, the method achieves significant accuracy gains and faster adaptation times for parametric partial differential equations. These findings advance operator learning by establishing neural solvers as efficient, local parametric models capable of handling complex physics simulations with minimal computational overhead.
Single-condition neural solvers encode transferable response spaces for parametric differential equations
arXiv:2609.15432v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
arXiv:2609.15432 [cs.LG]
(or arXiv:2609.15432v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.15432 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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