Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory
arXiv:2605.25509v1 Announce Type: cross Abstract: Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error
The proliferation of advanced AI techniques, particularly in generative models and flow matching, is enabling new approaches to complex scientific computing challenges.
This development allows for more efficient and accurate simulation and inverse problem solving in scientific and engineering fields, potentially accelerating research and development cycles.
Traditional computational simulation and inverse problem-solving methods can now be augmented or replaced by AI-driven generative frameworks that operate with sparse data.
- · Scientific Computing
- · AI/ML Research
- · Engineering Design
- · Materials Science
- · Traditional Simulation Software Vendors (slow to adapt)
- · Manual PDE Analysis
Improved efficiency and accuracy in solving Partial Differential Equations for various scientific and industrial applications.
Reduced computational cost and time for complex simulations, leading to faster innovation in design and discovery.
Democratization of advanced simulation capabilities, allowing more researchers and engineers to tackle previously intractable problems.
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Read at arXiv cs.LG