Deterministic Numeric Simulation and Surrogate Models with White and Black Machine Learning Methods: A Case Study on Direct Mappings

The approximation and emulation of first principles based deterministic models are important problems in many disciplines, like physical and natural sciences, as well as in engineering (industrial design, creation of digital twins and other tasks). Typically they involve complex systems, described by partial differential or integral equations which must be solved for a variety of space and time boundary conditions. Finding these solutions is usually costly in terms of both computational resources and time. Surrogate models are an effective way of building approximations that may replace the use of the compled/costly original models, expediting and speeding operations. Computational intelligence techniques have proven suitable for surrogating purposes and this paper explores the characterization of a relatively simple deterministic system described by a partial differential equation, using white as well as black box approaches for direct supervised mappings (inverse mappings are explored elsewhere). In addition, unsupervised methods are used for gaining insight into the properties of the input and output state spaces. White-box ML techniques exposed the nature of the inter-dependencies and the importance of the predictor variables. Individually, support vector regression outperformed all other models for the fixed-location, fixed time and also for the fixedlocation, time dependent scenario. However, performance-wise, the ensemble composed of white-box techniques outperformed the one integrated by black-box methods from the point of view of error and correlation measures.