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This paper establishes a dimension-efficient neural network approximation theory for fractional parabolic partial differential equations, focusing on solutions with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, the authors develop a maximal regularity theory that is dimension-independent, leveraging multiplication estimates and continuity methods to handle lower-order terms effectively. A significant finding is the derivation of $n^{-1/2}$ approximation bounds in mixed Sobolev norms, which highlights the limitations of uniform-in-time estimates for spectral Barron regularity under certain conditions.
Achieving dimension-independent approximation bounds for fractional parabolic PDEs could revolutionize how we model complex dynamical systems with neural networks.
We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.
