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.
翻译:最近有人提议将DeepONet作为学习非线性操作员在无限的Banach 空间间绘制非线性操作员的框架。 我们分析DeepONets, 并证明由此得出的近似和概括误差的估计数。 特别是, 我们扩展了DeepONets的通用近似属性, 以包括非compact 空间的可测量绘图。 通过将错误分解成编码、 近似和重建错误, 我们证明在总误差上, 与常态操作员的光谱性衰减特性有关, 与基本测量相联。 我们得到的几乎是最佳的误差界限, 与非常一般的直系重建器和随机的传感器位置, 以及普通误差的误差。 我们用四个非线性操作员的准近近近近似例子展示了我们的总框架, 即非线性硬性硬性ODE, 具有可变系数的非线性pariptial pal和超直线性PDEs。 Deepschitesitz 操作员的任意性 Lipschitive Lipschite 操作员与 $lonlontalatealate 等操作员的近近似接近值,,,, 。我们说, 可以用“crecial liversaltium liversal netline listal listal compeutebility netm licomm licomm lible ” licomm sal ex sal ex ex ex ex ex ex ex ex ex ex ex ex ex exbel ex ex, exm ex, exublex ex sal ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex exubal ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex