Deep Operator Networks (DeepOnets) have revolutionized the domain of
scientific machine learning for the solution of the inverse problem for
dynamical systems. However, their implementation necessitates optimizing a
high-dimensional space of parameters and hyperparameters. This fact, along with
the requirement of substantial computational resources, poses a barrier to
achieving high numerical accuracy. Here, inpsired by DeepONets and to address
the above challenges, we present Random Projection-based Operator Networks
(RandONets): shallow networks with random projections that learn linear and
nonlinear operators. The implementation of RandONets involves: (a)
incorporating random bases, thus enabling the use of shallow neural networks
with a single hidden layer, where the only unknowns are the output weights of
the network's weighted inner product; this reduces dramatically the
dimensionality of the parameter space; and, based on this, (b) using
established least-squares solvers (e.g., Tikhonov regularization and
preconditioned QR decomposition) that offer superior numerical approximation
properties compared to other optimization techniques used in deep-learning. In
this work, we prove the universal approximation accuracy of RandONets for
approximating nonlinear operators and demonstrate their efficiency in
approximating linear nonlinear evolution operators (right-hand-sides (RHS))
with a focus on PDEs. We show, that for this particular task, RandONets
outperform, both in terms of numerical approximation accuracy and computational
cost, the ``vanilla" DeepOnets.

通过随机投影算子网络（RandONets）解决动态系统的逆问题，实现线性和非线性算子的学习，降低参数空间维度，证明了 RandONets 对非线性算子和线性非线性演化算子的普适逼近准确性，相对于传统深度算子网络 DeepONets 具有更高的数值逼近精度和计算效率。