Laplace Neural Operator

What kind of problems is it mostly used for? Please describe.

This method is used for both ODE and PDE solving. Its been shown to have better performance than FNOs and GRU-based RNNs. It can handle non-periodic signals better than the others

Describe the algorithm you’d like
Laplace neural operators use the Pole-Residue method of the Laplace transformation, operating on the input function to produce the target output function. It does so by mapping the function into a high-dimensional Laplace space using learnable Pole and Residue tensors, extracting global features of the input via modes, and returning to the original space through an inverse Laplace operation.

Other implementations to know about

• original implementation
• neuraloperator library implementation (WIP)

References
Cao, Q., Goswami, S. & Karniadakis, G.E. Laplace neural operator for solving differential equations. Nat Mach Intell 6, 631–640 (2024). https://doi.org/10.1038/s42256-024-00844-4

Start with the linked original Laplace Neural Operator implementation, the neuraloperator pull request, and the cited paper to understand the pole-residue algorithm and expected inputs and outputs. Then identify the appropriate integration point in NeuralOperators.jl; done means a Julia implementation that supports the described ODE and PDE operator use cases and has project-appropriate validation.