boundary-compatible fno

Fourier Neural Operators (FNOs) show promise for PDE modeling but suffer from insufficient theoretical grounding, implementation ambiguities, and sensitivity to discretization—undermining reliability in practice. Method: We systematically formalize FNO’s operator-theoretic foundations and clarify its intrinsic frequency-domain modeling mechanism, correcting the common misconception that “frequency domain” equates to mere Fourier transforms. We propose a discretization-invariant implementation paradigm based on learnable frequency-domain filters. Leveraging NeuralOperator 2.0.0, we deliver a modular, theory-aligned open-source implementation supporting multi-resolution generalization and global correlation modeling. Contribution/Results: Experiments demonstrate substantial improvements in stability, generalization, and deployment efficiency across challenging PDE systems—including Navier–Stokes and Darcy flow—establishing a robust, accessible infrastructure for operator learning in scientific machine learning.

This work addresses aliasing errors introduced in the discrete implementation of Fourier Neural Operators (FNOs), which remain theoretically unquantified despite their practical significance. Specifically, the discrepancy between the continuous FNO formulation and its grid-based discretization has not been systematically characterized, and isolating this discretization error from other sources—such as approximation and optimization errors—remains challenging. Method: Leveraging tools from Fourier analysis, numerical functional analysis, and FFT theory, we derive an explicit algebraic convergence rate for the discretization error with respect to grid resolution and establish its quantitative dependence on the Sobolev regularity of the input function. Contribution/Results: We provide the first verifiable upper bound on this error, revealing intrinsic trade-offs among resolution, input smoothness, and model stability. Numerical experiments confirm the theoretical prediction: for inputs with higher Sobolev regularity, the error decays significantly under mesh refinement—thereby furnishing a rigorous foundation for reliable discrete FNO design.

This study addresses the limitation of Fourier Neural Operators in modeling non-periodic and local fine-scale structures due to their fixed spectral bases and hard truncation. To overcome this, we propose SAFDNO, which introduces stochastic adaptive Fourier decomposition theory to construct input-adaptive Takenaka-Malmquist orthogonal systems that replace predefined Fourier modes. Furthermore, a neural pole predictor is designed to amortize the stochastic selection process, enabling learned filtering on the analytic branch coefficients within Hardy spaces. Experimental results demonstrate that SAFDNO achieves superior performance across nine PDE benchmarks, significantly improving solution accuracy for equations such as Darcy flow and enhancing zero-shot super-resolution capabilities.

This work addresses the limitations of Fourier neural operators in terms of generalization accuracy and computational efficiency for mappings in function spaces. To overcome these challenges, the authors propose an efficient architecture that uniquely integrates rank-1 lattice sampling with hyperbolic cross truncation in the frequency domain. This combination reduces high-dimensional Fourier transforms to one-dimensional fast Fourier transforms and introduces a structured lattice-based training set in parameter space. The resulting method substantially decreases the number of model parameters, spatial sampling points, and required training samples. Evaluated on elliptic partial differential equations over toroidal domains, the approach achieves higher approximation accuracy while maintaining lower computational complexity compared to existing methods.

This study investigates the practical performance of Fourier Neural Operators (FNOs) in cross-resolution generalization and critically examines the validity of their assumed resolution equivariance. By comparing direct high-resolution inference against low-resolution inference followed by Fourier zero-padding upsampling, and complemented with inter-layer spectral analysis, the work reveals that FNOs concentrate spectral energy toward low frequencies in intermediate layers, while high-frequency details are predominantly reconstructed in later nonlinear stages. The authors identify nonlinear aliasing as a key factor impeding zero-shot resolution equivariance. Notably, on the Darcy flow benchmark, direct high-resolution inference does not consistently outperform the upsampling baseline, prompting the proposal of a simple yet effective cross-resolution evaluation protocol.

This work addresses the limitation of existing Fourier Neural Operators (FNOs) in effectively modeling structured couplings among Fourier modes in nonlinear partial differential equations. To overcome this, the authors propose a Higher-Order Spectral Convolution mechanism (HO-FNO), which explicitly introduces multilinear mode mixing in the frequency domain. This approach generalizes FNO’s diagonal modulation to higher-order nonlinear interactions, embedding inductive biases aligned with the dynamics of nonlinear PDEs. Empirical results demonstrate that HO-FNO significantly outperforms current spectral neural operators across multiple benchmarks: a single-layer HO-FNO surpasses a 16-layer FNO in highly nonlinear regimes and matches or exceeds the performance of advanced Transformers and state-space models.

Existing neural operators struggle to efficiently model parametric and coupled partial differential equations (PDEs). This work addresses this limitation by extending the Fourier Neural Operator (FNO) with minimal architectural modifications: it introduces a hypernetwork-driven, parameter-aware modulation mechanism to condition the operator on physical parameters, and systematically designs an operator structure for coupled PDEs that balances shared representations with cross-variable interactions. The resulting approach significantly improves modeling accuracy while preserving computational efficiency. On benchmark problems including capacitively coupled plasma and the Gray–Scott system, the method reduces prediction errors by 55%–72% compared to strong baselines.

This work addresses the challenge of balancing accuracy and computational efficiency in modeling two-dimensional Rayleigh–Bénard convection by proposing a lightweight Fourier Neural Operator (FNO) architecture based on temporal increment prediction. Instead of predicting the full solution directly, the model forecasts the state increment, substantially reducing its complexity. The resulting network contains only 314k parameters (1.26 MB) and achieves a single inference time of 7 milliseconds. While maintaining accuracy comparable to existing methods, the approach significantly lowers computational overhead, offering a novel pathway toward efficient, high-fidelity simulations in fluid dynamics.

This work addresses the challenge that Fourier Neural Operators (FNOs) struggle to generalize across variable geometries and independent discretizations by introducing a deterministic framework. The physical domain is embedded into a fixed hypercube, with geometry represented via a signed distance function. Input and solution fields are extended into an ambient domain and processed by an FNO on a unified, non-uniform Cartesian latent grid, after which they are interpolated back to the target mesh and restricted to the physical domain. This approach requires no trainable graph networks, point clouds, or geometric decoding modules, fully decoupling geometry handling from the optimization pipeline and enabling unified modeling of arbitrary domains. Evaluated on 2D/3D nonlinear Poisson and convection–reaction–diffusion problems, the method achieves relative L² errors of 0.32%–0.77%; when used as an initial field in CFD simulations, it reduces pseudo-time iterations by 44% on average, accelerates URANS physical time marching by 18.52%–27.51%, and shortens cross-condition DNS guidance intervals by 23.47%–48.21%.

This study addresses the limitations of Fourier Neural Operators (FNOs) in learning high-frequency components due to frequency truncation, as well as the challenges in coordinate encoding design. To this end, we propose the CAFE+FNO framework, which introduces a novel CAFE+ mechanism that fuses Fourier-Chebyshev features via the Hadamard product to enhance frequency-domain interactions. By integrating implicit neural representations, parallel affine branches, and shared-kernel MLPs, the method generates Fourier kernels through explicit feature composition, enabling efficient PDE solving with model parameters independent of the number of modes. Experiments across five PDE benchmarks demonstrate that the proposed approach significantly outperforms existing FNO variants, fully validating its advantages in bandwidth learnability.