arXiv:2606.01244v1 Announce Type: cross Abstract: We study operator learning using encoder--decoder neural networks. Inspired by the function-space theory of neural networks, we introduce a variation space as an infinite-dimensional structural class for nonlinear operators. This space is defined through vector-valued measures directly on the input and output spaces. For operators in this space, we establish approximation bounds for encoder--decoder two-layer networks in the Bochner $L^q$ norm. The resulting error bound decomposes into the input encoding error, the output encoding error, and a
The continuous drive for more efficient and robust AI models, particularly in operator learning for complex systems, necessitates new theoretical and architectural innovations.
This research provides a more rigorous theoretical foundation for neural operators, which are crucial for AI applications needing to learn mappings between infinite-dimensional function spaces, impacting scientific computing and AI agent development.
The proposed 'variation space' offers a novel theoretical framework to understand and improve neural operator performance, potentially leading to more accurate and generalizable AI models for complex tasks.
- · AI researchers and developers
- · Companies investing in scientific AI
- · Sectors requiring complex function approximation
- · Traditional numerical methods (in long-term)
Improved efficiency and approximation capabilities of AI models for operator learning.
Faster development and deployment of AI agents capable of understanding and manipulating complex physical or abstract systems.
Acceleration of scientific discovery and engineering design through highly efficient and accurate AI-driven simulations and control systems.
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Read at arXiv cs.LG