Innovation in Mathematical Physics
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Abstract
Fourier Neural Operators (FNOs) have emerged as a significant breakthrough in solving mathematical physics equations, such as the heat, Navier-Stokes, and Kermack-McKendrick equations. By learning to map data in the frequency domain rather than computing over traditional mesh grids, FNOs circumvent the intensive recalculations typically required across varying grid sizes and parameter regimes. However, FNOs often struggle with highly non-linear boundary conditions. In this work, we propose a modified FNO architecture that resolves this bottleneck.] We demonstrate the approach maintains accuracy across modestly different parameters while accelerating computation times. This framework is particularly advantageous for iterative optimization tasks, such as aerodynamic shape design, where repeated simulations are required to converge on an optimal configuration. Results show a100x speedup over classical solvers without loss of fidelity.
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Copyright (c) 2026 Lewis TG.
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