Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments
This addresses the problem of efficiently modeling non-Markovian processes for researchers and practitioners in stochastic systems, representing a novel method for a known bottleneck.
The paper tackled learning dynamics of stochastic systems by introducing the mirror-padded Fourier neural operator (MFNO), which extends standard FNOs to handle non-periodic inputs and approximates solutions of path-dependent stochastic differential equations and fractional Brownian motions with arbitrary accuracy. Empirically, MFNO showed strong resolution generalization, outperformed or matched baselines like LSTMs and DeepONet, and offered faster sample path generation than classical numerical schemes.
This paper introduces an operator-based neural network, the mirror-padded Fourier neural operator (MFNO), designed to learn the dynamics of stochastic systems. MFNO extends the standard Fourier neural operator (FNO) by incorporating mirror padding, enabling it to handle non-periodic inputs. We rigorously prove that MFNOs can approximate solutions of path-dependent stochastic differential equations and Lipschitz transformations of fractional Brownian motions to an arbitrary degree of accuracy. Our theoretical analysis builds on Wong--Zakai type theorems and various approximation techniques. Empirically, the MFNO exhibits strong resolution generalization--a property rarely seen in standard architectures such as LSTMs, TCNs, and DeepONet. Furthermore, our model achieves performance that is comparable or superior to these baselines while offering significantly faster sample path generation than classical numerical schemes.