Research map / Probabilistic and scientific modeling / Uncertainty and Gaussian processes
Infinite Neural Operators: Gaussian processes on functions
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Shows that arbitrary-depth neural operators with Gaussian-distributed kernels converge to function-valued Gaussian processes as width grows, derives their covariance functions for FNO and a toroidal Matérn operator, and uses them for GP regression.
Problem
Whether infinitely wide neural operators converge to Gaussian processes over functions, and how to characterize their covariance, given that function spaces are infinite-dimensional.
Approach
Extends the infinite-width BNN-to-GP correspondence to neural operators by placing i.i.d. GP priors on integral kernels, using a compositionality lemma (via the strong law of large numbers) to compose covariances layer by layer, and deriving closed-form covariances for band-limited FNO kernels and a toroidal Matérn kernel.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase hidden channel width | finite-width FNO · initialization distribution | total variation distance | Helped |
| increase the number of Monte Carlo samples | neural operator layer variance estimate · initialization variance validation | variance estimation agreement | Helped |
| vary the kernel band-limit (ablation) | FNO and ∞-FNO · synthetic regression | test L2 loss | Mixed |
Limitations
- Covariance computation scales cubically in evaluation grid size and number of training functions.
- Theory covers only the distribution at initialization; posterior comparison to SGD-trained networks is not theoretically justified.
- Results restricted to the flat torus with band-limited kernels.
Experimental setup
Datasets: synthetic operator data from randomly initialized FNO 1D Burgers' equation dataset (nu=0.002)
Benchmarks: 1D Burgers' equation (nu=0.002) end-state prediction synthetic FNO-generated regression
Metrics: L2 test loss total variation distance to Gaussian variance agreement with Monte Carlo
Baselines: FNO trained with Adam
Builds on: Fourier neural operator neural network Gaussian process limit operator-valued kernels
Closest papers by meaning
- Linearization Turns Neural Operators into Function-Valued Gaussian Processes (ICML 2025 · similarity 0.91)
- Optimization for Neural Operators can Benefit from Width (ICML 2025 · similarity 0.87)
- Neural Operators with Localized Integral and Differential Kernels (ICML 2024 · similarity 0.87)
- Guaranteed Approximation Bounds for Mixed-Precision Neural Operators (ICLR 2024 · similarity 0.87)
- Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains (ICML 2024 · similarity 0.87)
- Thompson Sampling in Function Spaces via Neural Operators (NeurIPS 2025 · similarity 0.86)
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