Solving Stochastic Differential and Integral Equations Using Neural Network Method

Artificial intelligence and nearly all its subfields include machine learning and deep learning in operations with the closings being a vital aspect across disciplines including solving mathematical problems. The presentation of this work proposes a novel approach to the computation of numerical solutions to ODEs with certain architectures of deep learning models namely single-layer CNN with logistic sigmoid activation. It uses a control variable approach which facilitates manipulation of hyperparameters and the optimizer's choice. The first goal is to develop CNN to solve ODEs with high precision powered by the Adam optimizer and backpropagation method. The extensive experiments include bespoke loss functions that capture ODE features, and analysis of the effects of different structures of neural networks on solutions. Various outcomes, such as the first application of the CNN-based approach, the assessment of adaptations through control variables, and the exploration of the effects exerted by the network structure on ODE solutions. This research find its application in the use of neural networks in solving mathematical problems particularly ODEs; finding a new way of solving them without consuming a lot of time. This paper has offered grounds for improved mathematical problem-solving and established the efficacy of CNNs and counteractive loss functions. The consequences have been carried out to numerous scientific and engineering fields and are the possibility of enhanced calculation accuracy and efficacy of ODEs in numerous practice areas. This paper presents the advantages of using convolutional neural networks (CNNs) to solve differential equations (ODEs), particularly with reference to the use of loss functions to capture important ODE features. This method, which incorporates traditional mathematical methods, reduces computational time by an average of 30% and improves accuracy by an average of 25%. Applications range from dynamic systems modeling in engineering to predictive analysis in biological systems and business forecasting. These results confirm the power and versatility of CNNs in solving mathematical problems.