This paper presents a deep learning-based numerical algorithm for solving path-dependent partial differential equations (PDEs) that arise in rough volatility models. The authors build on the connection between PDEs and backward stochastic differential equations (BSDEs), as previously explored by Bayer, Qiu, and Yao, and introduce random-weight neural networks (RWNNs) to approximate the solution. In RWNNs, the hidden layers are randomly initialized and fixed, while only the final linear layer is trained, reducing the optimization to a convex least-squares regression problem. This approach significantly accelerates training compared to traditional deep neural networks while maintaining comparable accuracy.

The algorithm is applied to both Markovian and non-Markovian settings, with a focus on rough volatility models such as the rough Bergomi model. The authors provide a convergence analysis, deriving error bounds in terms of the number of hidden nodes. Numerical experiments demonstrate that the RWNN-based method achieves efficient and accurate pricing for options under rough volatility, though it still requires Monte Carlo path simulation. The paper highlights that the additional complexity of deep neural networks may be unnecessary when precise error bounds are desired, and the randomized least-squares formulation allows for efficient implementation using tensor operations.