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This research introduces an auxiliary function dimension to reformulate Neural Operators in $d+1$ dimensions, offering a computationally efficient alternative to brute-force embedding scaling for learning mappings between function spaces. By leveraging Fourier-based operators across physical and auxiliary domains, the model achieves superior $L_2$ error performance on benchmarks ranging from 1D heat equations to 3D Rayleigh-Taylor instability while maintaining robustness under mixed-resolution training. The approach demonstrates strong zero-shot generalization to unseen temporal regimes, suggesting a promising direction for optimizing approximation accuracy without increasing computational budget.
Reformulating Neural Operators in $d+1$ Dimensions for Embedding Evolution
arXiv:2505.11766v4 Announce Type: replace-cross Abstract: Neural Operators (NOs) are powerful architectures for learning mappings between function spaces. While most advances focus on refining kernel parameterizations over the $d$-dimensional physical domain, the evolution of lifted embeddings remains underexplored, which often drives models toward computationally expensive embedding-scaling designs to improve approximation. In this paper, we introduce an auxiliary function dimension that models embedding evolution in operator form, thereby reformulating the NO pipeline in $d+1$ dimensions. We instantiate this framework via Fourier-based operators acting jointly on the physical and auxiliary domains, yielding a basis-diversified auxiliary evolution module as an alternative to brute-force embedding scaling. Across more than ten increasingly challenging benchmarks, ranging from the 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability, our model consistently achieves the lowest relative $L_2$ error among the evaluated baselines. Crucially, this advantage is empirically supported by (1) controlled budget-aware comparisons against scaled and ablated baselines; (2) robustness under mixed-resolution training and super-resolution inference; and (3) zero-shot generalization to unseen temporal regimes. In addition, we present a broader set of design choices for lifting and recovery operators, demonstrating their impact on our model's predictive performance.
Abstract:Neural Operators (NOs) are powerful architectures for learning mappings between function spaces. While most advances focus on refining kernel parameterizations over the $d$-dimensional physical domain, the evolution of lifted embeddings remains underexplored, which often drives models toward computationally expensive embedding-scaling designs to improve approximation. In this paper, we introduce an auxiliary function dimension that models embedding evolution in operator form, thereby reformulating the NO pipeline in $d+1$ dimensions. We instantiate this framework via Fourier-based operators acting jointly on the physical and auxiliary domains, yielding a basis-diversified auxiliary evolution module as an alternative to brute-force embedding scaling. Across more than ten increasingly challenging benchmarks, ranging from the 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability, our model consistently achieves the lowest relative $L_2$ error among the evaluated baselines. Crucially, this advantage is empirically supported by (1) controlled budget-aware comparisons against scaled and ablated baselines; (2) robustness under mixed-resolution training and super-resolution inference; and (3) zero-shot generalization to unseen temporal regimes. In addition, we present a broader set of design choices for lifting and recovery operators, demonstrating their impact on our model's predictive performance.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Quantum Physics (quant-ph)
arXiv:2505.11766 [cs.LG]
(or arXiv:2505.11766v4 [cs.LG] for this version)
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