RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators
<|reference_start|>RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators: 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.<|reference_end|>
@article{fabiani2024randonet:,
    title={RandONet: Shallow-Networks with Random Projections for learning linear
  and nonlinear operators},
    author={Gianluca Fabiani, Ioannis G. Kevrekidis, Constantinos Siettos,
  Athanasios N. Yannacopoulos},
    journal={Journal of Computational Physics, 520, 113433 (2025)},
    year={2024},
    doi={10.1016/j.jcp.2024.113433},
    archivePrefix={arXiv},
    eprint={2406.05470},
    primaryClass={cs.LG cs.NA math.DS math.NA}
}