Chemical process industries face challenges in monitoring and controlling complex operations. Current `visualization' techniques require extensive data and computational efforts, which may not always be feasible. Moreover, these techniques often lack detailed insights into the underlying physics. This thesis explores an a priori integral approach that can reconstruct scalar or vector fields using sparse sensor measurements. The approach is conceptually built on the mathematical constraints associated with Helmholtz-Hodge Decomposition together with the physical laws. To illustrate this, temperature field reconstructions are considered in steady heat transfer systems, including scenarios with either heat generation or forced convection, using discrete data obtained from flow-following sensors.

A generalized framework is developed using hypothetical heat sources (potentials), with parameters of the heat potentials being determined from the values of the temperature field measured at limited discrete points. Infinitely many reconstructed solutions are possible and the arrangement, population, intensity, and size of the hypothetical heat potentials are the issues of interest that influence the reconstructed solution. Two concrete possibilities are presented for simplification (linearization) by limiting the issues associated with these potentials. The optimal values of unknowns are determined using sparse sensor measurements in a linear system of equations, with the help of `training’. The framework-assisted reconstructed fields demonstrate accurate predictions of uniform and smooth temperature distribution, while utilizing only a small number of sensor measurements, and minimal computational effort. This validates the effectiveness of an integral approach.

In more complex situations, like locally uneven fields or sharp convective currents, the framework-assisted reconstructions focus primarily on the dominant phenomena and do not capture specific (or, local) characteristics. This highlights the inherent limitations associated with the simplification of a non-linear problem. Potential improvements regarding the treatment of issues associated with heat potentials are suggested for developing a more versatile framework.

The performance of the source framework developed in this work, based on an integral approach, is compared with the Hidden Fluid Mechanics algorithm, a recently developed physics-informed neural network framework, based on a differential approach. This comparison highlights the strength of the source frameworks and the integral approach to `visualize' simple and smooth domains, in terms of computational expense and accuracy, particularly when dealing with a limited number of measurements. Integrating physics-constrained field functions, developed in this work, into a neural network architecture can present an intriguing avenue for framework optimization. Additionally, gradual enhancements in domain complexity can be explored to expand the applicability of the framework.