arXiv:2606.06046v1 Announce Type: cross Abstract: We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matching pursuit (OMP), to substantially reduce the required sample size compared with a previously considered cubature-based approach. We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourie
This research is part of the ongoing effort to enhance the computational efficiency and accuracy of AI models applied to complex scientific problems.
Improved methods for solving PDEs can unlock advancements in scientific computing, engineering, and various AI applications that rely on simulating physical systems.
The ability to reduce sample sizes in solving PDEs through sparse approximation can accelerate research and development in fields requiring extensive simulations.
- · AI/ML researchers
- · Scientific computing sector
- · Engineering industries
- · Drug discovery & materials science
- · Traditional high-computational methods
- · Sectors reliant on slow simulation cycles
More efficient and accurate simulation of complex physical phenomena becomes possible.
This could lead to faster innovation cycles in areas like climate modeling, aerospace design, and pharmaceutical development.
Reduced computational costs might democratize access to advanced simulation capabilities, fostering broader scientific discovery.
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Read at arXiv cs.LG