Backpropagation within neural networks leverages a fundamental element of automatic differentiation, which is referred to as the reverse mode differentiation, or vector Jacobian Product (VJP) or, in the context of differential geometry, known as the pull-back process. The computation of gradient is important as update of neural network parameters is performed using gradient descent method. In this study, we present a genric randomized method, which updates the parameters of neural networks by using directional derivatives of loss functions computed efficiently by using forward mode AD or Jacobian vector Product (JVP). These JVP are computed along the random directions sampled from different probability distributions e.g., Bernoulli, Normal, Wigner, Laplace and Uniform distributions. The computation of gradient is performed during the forward pass of the neural network. We also present a rigorous analysis of the presented methods providing the rate of convergence along with the computational experiments deployed in scientific Machine learning in particular physics-informed neural networks and Deep Operator Networks.

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