Scattering with Neural Operators (2308.14789v2)

Abstract: Recent advances in machine learning establish the ability of certain neural-network architectures called neural operators to approximate maps between function spaces. Motivated by a prospect of employing them in fundamental physics, we examine applications to scattering processes in quantum mechanics. We use an iterated variant of Fourier neural operators to learn the physics of Schr\"odinger operators, which map from the space of initial wave functions and potentials to the final wave functions. These deep operator learning ideas are put to test in two concrete problems: a neural operator predicting the time evolution of a wave packet scattering off a central potential in $1+1$ dimensions, and the double-slit experiment in $2+1$ dimensions. At inference, neural operators can become orders of magnitude more efficient compared to traditional finite-difference solvers.

• The paper introduces a novel iterated Fourier neural operator framework to model scattering in quantum mechanics.
• It validates the approach with numerical experiments that accurately simulate wave packet scattering and the double-slit experiment while maintaining unitarity.
• The results suggest that neural operators can efficiently generalize and potentially replace finite-difference methods for long-term quantum dynamics.

Scattering with Neural Operators: A Formal Analysis

This paper, titled "Scattering with Neural Operators," presents a methodological exploration into utilizing neural operators to model scattering processes in quantum mechanics. The research implements an iterated variant of Fourier neural operators to approximate the physics of Schrödinger operators, focusing particularly on two scenarios: wave packet scattering in a $(1+1)$-dimensional setup and the classic two-dimensional double-slit experiment. This exploration seeks to validate the potential for neural operators in providing computationally efficient solutions to problems traditionally solved by finite-difference methods.

The primary structure of this research leverages the capability of neural operators to form mappings between function spaces, thus encoding the time evolution intrinsic to Schrödinger operators. Neural operators, in contrast to traditional neural networks, target more complex interactions by approximating non-linear operators that involve infinite-dimensional inputs. The particular architecture examined in this paper is based on Fourier neural operators (FNOs), chosen for their integration with fast Fourier transforms to empirically test their applicability in solving quantum mechanical equations.

Key insights from the paper arise from the numerical experiments demonstrating the efficacy of the proposed neural operator framework. Training is conducted on synthetic datasets generated by varying Gaussian noise added to potentials and initial states, divorced from any specific physics-based setups, thus emphasizing the model's capacity to generalize. Numerical experiments are extended beyond the training regime by using wave packet and double-slit tests as benchmarks for out-of-distribution performance, showing that the trained neural operator adequately models complex interference patterns with acceptable errors.

Among the noteworthy claims, one numerical result highlights the neural operators' ability to maintain high calculation fidelity across significant time evolutions. For example, in the wave packet scattering model, even though prediction errors naturally compound when time steps are iterated, the final results demonstrate adequate accuracy in representing wave function evolution, with unitarity maintained. This result suggests a promising extrapolation of neural operators to model long-term dynamics traditionally handled by computationally intensive methods.

The implications of deploying neural operators in quantum mechanics extend beyond mere computational advantages. Theoretically, neural operators offer a paradigm shift in representing Hamiltonians and S-matrix calculations, potentially addressing the burgeoning complexity challenges faced in asymmetric, high-energy particle dynamics simulations. Practically, the computational efficiency surfaced, particularly in scaling to higher lattice resolutions without prohibitive memory constraints, indicates a path toward real-time, large-scale quantum dynamics modeling applications.

In conclusion, the research pioneers a conceptual and practical demonstration of neural operators' utility to simulate quantum mechanical scattering processes efficiently. Future work may delve into tailored training datasets sourced from quantum mechanics to further hone out-of-distribution error strategies, ultimately paving the way for neural operators to augment traditional quantum computation methodologies in both academic research and applied engineering contexts.