This paper presents a residual-informed machine learning approach for replacing algebraic loops in equation-based Modelica models with neural network surrogates. A feedforward neural network is trained using the residual (error) of the algebraic loop directly in its loss function, eliminating the need for a supervised dataset. This training strategy also resolves the issue of ambiguous solutions, allowing the surrogate to converge to a consistent solution rather than averaging multiple valid ones. Applied to the large-scale IEEE 14-Bus system, our method achieves a 60% reduction in simulation time compared to conventional simulations, while maintaining the same level of accuracy through error control mechanisms.

One of the core concepts in science, and something that happens intuitively in every-day dynamic systems modeling, is the combination of models or methods. Especially in dynamical systems modeling, often two or more structures are combined to obtain a more powerful or efficient architecture regarding a specific application (area). Further, even physical simulations are combined with machine learning architectures, to increase prediction accuracy or optimize the computational performance. In this work, we shortly discuss, which types of models are usually combined and propose a model interface that is capable of expressing a width variety of mixed algebraic, discrete and differential equation based models. Further, we examine different established, as well as new ways of combining these models from a system theoretical point of view and highlight two challenges - algebraic loops and local event affect functions in discontinuous models - that require a special approach. Finally, we propose a new wildcard topology, that is capable of describing the generic connection between two combined models in an easy to interpret fashion that can be learned as part of a gradient based optimization procedure. The contributions of this paper are highlighted at a proof of concept: Different connection topologies between two models are learned, interpreted and compared applying the proposed methodology and software implementation.