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Backpropagation within neural networks leverages a fundamental element ofautomatic differentiation, which is referred to as the reverse modedifferentiation, or vector Jacobian Product (VJP) or, in the context ofdifferential geometry, known as the pull-back process. The computation ofgradient is important as update of neural network parameters is performed usinggradient descent method. In this study, we present a genric randomized method,which updates the parameters of neural networks by using directionalderivatives of loss functions computed efficiently by using forward mode AD orJacobian vector Product (JVP). These JVP are computed along the randomdirections sampled from different probability distributions e.g., Bernoulli,Normal, Wigner, Laplace and Uniform distributions. The computation of gradientis performed during the forward pass of the neural network. We also present arigorous analysis of the presented methods providing the rate of convergencealong with the computational experiments deployed in scientific Machinelearning in particular physics-informed neural networks and Deep OperatorNetworks.

Randomized Forward Mode of Automatic Differentiation for Optimization Algorithms