Theory of Functional Connections Extended to Continuous Integral Constraints

This study extends the Theory of Functional Connections, previously applied to constraints specified at discrete points, to encompass continuous integral constraints of the form ∫x0xff(x,t)dx=I(t), where I(t) can be a constant, a prescribed function, or an unknown function to be estimated through optimization. The framework of continuous integral constraints is developed within the context of initial value problems (IVP) and boundary value problems (BVP). To demonstrate the effectiveness of this analytical approach, examples validate the method and highlight distinctions between satisfying continuous integral constraints via simple interpolation versus functional interpolation. A limitation of the proposed approach is the inability to inherently enforce inequality constraints, such as the positivity constraint f(x,t)≥0, for modeling probability density functions in classical mechanics. Despite this, numerical experiments on boundary-value problems rarely result in negative values, indicating that the issue occurs infrequently. However, a mitigation strategy based on non-negative least-squares methods combined with Bernstein polynomials is proposed to address these rare cases. This approach is validated through an additional numerical test, demonstrating its efficacy in ensuring nonnegativity when required.