Neural Scaling Laws of Transformer Operator Network

arXiv:2609.33533v1 Announce Type: cross Abstract: Transformers have emerged as powerful architectures for learning solution operators of physical systems. Empirically the prediction error has been observed to decrease when the data size and model size increase, suggesting neural scaling behavior. Yet a theoretical understanding of such scaling laws for transformer-based operator learning remains limited. In this work, we develop a theoretical framework for characterizing the approximation and generalization errors of transformer-based operator learning. Our analysis builds on a local-to-global approximation principle that is naturally aligned with the softmax attention mechanism and yields discretization-invariant output functions. On approximation theory, we derive a universal approximation error of transformer-based operator learning for H\"older-regular operators. On generalization theory, we establish a power scaling law between the prediction error and the training data size. The rate of convergence represented by the scaling exponent explicitly reflects the dimensions of the input and output domains, the regularity of the underlying functions and operators, and crucially, the intrinsic dimension of the input function class. By exploiting this intrinsic low-dimensional structure, our analysis yields a power-law generalization rate for operator learning, in contrast to the logarithmic-type power-law rates appearing in existing analyses of operator learning with feedforward neural networks. Numerical experiments validate the predicted power-law scaling and confirm that the convergence rate varies systematically with the intrinsic dimension of the input function class.

Abstract:Transformers have emerged as powerful architectures for learning solution operators of physical systems. Empirically the prediction error has been observed to decrease when the data size and model size increase, suggesting neural scaling behavior. Yet a theoretical understanding of such scaling laws for transformer-based operator learning remains limited. In this work, we develop a theoretical framework for characterizing the approximation and generalization errors of transformer-based operator learning. Our analysis builds on a local-to-global approximation principle that is naturally aligned with the softmax attention mechanism and yields discretization-invariant output functions. On approximation theory, we derive a universal approximation error of transformer-based operator learning for Hölder-regular operators. On generalization theory, we establish a power scaling law between the prediction error and the training data size. The rate of convergence represented by the scaling exponent explicitly reflects the dimensions of the input and output domains, the regularity of the underlying functions and operators, and crucially, the intrinsic dimension of the input function class. By exploiting this intrinsic low-dimensional structure, our analysis yields a power-law generalization rate for operator learning, in contrast to the logarithmic-type power-law rates appearing in existing analyses of operator learning with feedforward neural networks. Numerical experiments validate the predicted power-law scaling and confirm that the convergence rate varies systematically with the intrinsic dimension of the input function class.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2609.33533 [stat.ML]

(or arXiv:2609.33533v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.33533 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

aiqbench analysis

Neural Scaling Laws of Transformer Operator Network

Related coverage