Let There Be Light: Reflection, Refraction and Scattering for Neural Operators
Authors: Keke Wu, Yixuan Zhang, Jingrun Chen
Organizations: Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China · School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei 230026, China · Suzhou Big Data & AI Research and Engineering Center, Suzhou 215123, China · School of Mathematical Science, Peking University, Beijing, 100871, China · School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.
This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation. The work analyzes key models, including DeepONet, PCANet, and the Fourier Neural Operator, highlighting their underlying representations, computational structures, and comparative performance. These architectures are demonstrated on three canonical PDE problems: the Poisson equation, a linear elasticity problem, and a hyperelasticity problem. To make the presentation self-contained, key foundational topics are introduced, including finite-dimensional representations of function spaces, singular-value decomposition, and sampling from infinite-dimensional function spaces. Beyond forward modeling, the review discusses the use of neural operators as surrogate models within a Bayesian inverse-problem framework, including prior specification, forward-map approximation, and posterior computation. The performance of the three neural-operator architectures is evaluated on in-distribution samples, out-of-distribution samples, and Bayesian inference tasks. The review also discusses challenges related to prediction accuracy and generalization, outlining emerging strategies such as residual-based error correction and multi-level training. The review concludes by positioning neural operators within broader scientific-computing workflows and by identifying directions for reliable, scalable operator learning.
Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations. Transformer-based neural operators are of particular interest, since attention can learn long-range dependencies in the computational domain. However, standard attention has two major limitations when applied to PDEs: it scales quadratically with the number of computational nodes, and it lacks an explicit bias toward local interactions. To address these issues, we introduce Local Linear Transformer (LLT) for PDE operator learning. The architecture combines linear global attention with local spatial mixing, and incorporates coordinate and geometry information. We evaluate LLT on several PDE problems, including elasticity, plasticity, airfoil flow, pipe flow, and Darcy flow. The reference data for these problems span finite-element, finite-volume, and finite-difference discretizations on structured and unstructured meshes. Compared with other neural-operator and transformer baselines from prior studies, LLT achieves competitive or lower relative L2 error across these problems. On matched structured discretizations, wall-clock time per training iteration is reduced by factors of 1.8 to 2.5 relative to Transolver. We also scale the approach and apply it to a three-dimensional car aerodynamics dataset with 32,186 unstructured mesh points per sample. Together, these results indicate that LLT provides an accurate and computationally efficient operator for PDE problems across discretizations, mesh types, and problem settings.
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.