Error estimates for DeepOnets: A deep learning framework in infinite dimensions (2102.09618v3)

Abstract: DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ODE, an elliptic PDE with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $\epsilon$ is argued to suffer from a "curse of dimensionality" (requiring a neural networks of exponential size in $1/\epsilon$), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $\epsilon$ with neural networks of size that can grow algebraically in $1/\epsilon$). Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.

• The paper provides a comprehensive theoretical analysis of error estimates for DeepONets, extending universal approximation theorems and decomposing total error into encoding, approximation, and reconstruction components.
• It demonstrates that DeepONets can potentially break the curse of dimensionality for certain classes of operators and provides explicit examples for various types of differential equations.
• The theoretical framework is illustrated with applications to nonlinear ODEs and various PDEs, showcasing DeepONets' capacity to handle complex functional mappings in science and engineering.

Analysis of the Error Estimates for DeepONets

The paper "Error estimates for DeepONets: A deep learning framework in infinite dimensions" presents a comprehensive theoretical analysis of Deep Operator Networks (DeepONets), which are designed to approximate nonlinear operators mapping between infinite-dimensional spaces. This framework has particular relevance in science and engineering, where such operators frequently arise in the paper of differential equations and their solutions.

DeepONets extend classical neural networks into a form suitable for operator approximation, where, unlike traditional setups that handle finite-dimensional vectors, the inputs and outputs are functions, often defined over complex domains. The paper establishes new theoretical foundations for understanding the error bounds associated with these networks and demonstrates that DeepONets can be effective even in the complex task of approximating mappings between infinite-dimensional spaces.

Contributions and Theoretical Foundations

Universal Approximation Theorem

The authors extend the universal approximation theorem for operator networks, originally introduced by Chen and Chen, to DeepONets, relaxing previously required continuity and compactness conditions. They prove that DeepONets can approximate any measurable operator to arbitrary precision with respect to a given probability measure on continuous function spaces.

Error Decomposition

A notable advancement in the paper is the rigorous decomposition of the total approximation error associated with DeepONets into three distinct components:

1. Encoding Error: Associated with the approximation of the infinite-dimensional input space to a finite-dimensional representation.
2. Approximation Error: Originating from the approximator, which is a neural network mapping finite-dimensional spaces.
3. Reconstruction Error: Linked to the process of reconstructing the output in infinite-dimensional space from its finite-dimensional approximation.

The paper provides bounds for each of these errors in terms of properties of the underlying probability measure (for encoding) and the spectral properties of a covariance operator (for reconstruction), ultimately offering a comprehensive framework for estimating the DeepONet's overall error.

Exploration of Spectral Properties

The spectral properties of covariance operators play a central role in analyzing the reconstruction error. The authors illustrate that even when the spectral decay of an input measure is rapid (e.g., exponential), a nonlinear operator can drastically change this property in the push-forward measure, emphasizing the complexity of operator mappings.

Practical Implications

Breakthrough in overcoming the Curse of Dimensionality

The analysis shows that under certain conditions, particularly when dealing with smooth outputs or holomorphic operators, DeepONets can break the curse of dimensionality. This is demonstrated through explicit examples where operator approximations achieve error rates that do not increase exponentially with dimension or precision, a common limitation in high-dimensional problems.

Applications to Differential Equations

The theoretical findings are illustrated through concrete examples involving differential equations, such as:

• Nonlinear ODEs (e.g., the gravity pendulum with external force)
• Elliptic PDEs with variable coefficients
• Nonlinear parabolic PDEs using reaction-diffusion models like the Allen-Cahn equation
• Hyperbolic PDEs exemplified by scalar conservation laws

These examples showcase DeepONets' capacity to handle the varying challenges posed by different types of differential equations, from ensuring smooth approximation to dealing with discontinuous solutions like those seen in shock waves.

Future Directions

The paper opens multiple avenues for future research, including refining the complexity estimates of DeepONets for specific applications, exploring alternative network architectures, and extending the results to higher-dimensional and more complex PDE systems. Moreover, as this framework becomes increasingly relevant in practical contexts, integrating these theoretical insights with empirical results will further enhance DeepONets' implementation in scientific and engineering applications.

Conclusion

In summary, this paper stands as an essential contribution to the field of deep learning in infinite-dimensional spaces, establishing a solid theoretical base for DeepONets. It not only enhances our understanding of the underlying mechanics of operator learning but also provides practical insights that can inform the design and training of advanced neural networks for complex functional mappings found in real-world applications.