Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps
arXiv:2606.29436v1 Announce Type: cross Abstract: We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout
The continuous advancements in AI and machine learning necessitate more robust and efficient methods for scientific computing and complex physical simulations, driving research into novel approaches like Fourier Neural Operators.
This research addresses a critical challenge in modeling complex physical systems with unknown or random parameters, which has broad applications across engineering, materials science, and environmental modeling.
The ability to learn obstacle-to-solution maps directly from sampled data, even with complex contact sets, improves the realism and utility of AI in predicting outcomes for systems with inherent randomness.
- · AI researchers in scientific computing
- · Engineering simulation software developers
- · Material scientists
- · Environmental modelers
- · Traditional numerical methods for complex variational inequalities
Improved accuracy and efficiency in simulating systems with random or unknown boundary conditions using AI.
Faster design cycles and optimized performance for products and systems where random elements are a factor, such as in material defect prediction or fluid dynamics.
Potential for AI-driven real-time control and adaptation in dynamic systems where obstacles or conditions change unpredictably.
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