A Physics Based Functionally Linked Constrained Neural Networks Framework for Solving Differential Equations

This paper presents a Physics-based Functionally Linked Constrained Neural Networks framework for the efficient and accurate solution of differential equations (DEs). The presented approach integrates the Theory of Functional Connections (TFC) with a Functional Link Neural Network (FLNN) to construct a constrained expression that exactly satisfies all prescribed initial and boundary conditions. By embedding the constrained directly into the solution, the framework eliminates the need to enforce these conditions through penalty terms during training. The free function component of the constrained expression is represented using an FLNN, which offers a computationally efficient alternative to deep neural network architectures. Model training is carried out by minimizing the mean squared error of the governing differential equation residual at selected collocation points. As a result, the presented framework maintains the physical consistency of the solution while reducing the computational burden associated with deep learning based approaches. To evaluate its effectiveness, this method is applied to several linear and nonlinear ordinary and partial differential equations. The obtained numerical results are compared with analytical solutions and existing Physics-Informed Neural Network (PINN) models. The comparisons demonstrate that the framework achieves higher accuracy, faster convergence, and significantly reduced training time while requiring fewer trainable parameters. Overall, this framework offers a fast, accurate, and efficient approach for solving differential equations in scientific and engineering applications.