Generating Stable and Metastable Critical Points in Uncertain Systems via Flow‐Based Models

ABSTRACT This work proposes the use of conditional flow‐based generative models to learn an approximation of the distribution of the critical points of a cost function. This approximation is used to incrementally identify all critical points, in the feasible domain of said function, by iteratively alternating the sampling of the distribution and the retraining of the model with the newly discovered points. This paper will focus, in particular, on the identification and conditional generation of all local minima in the case in which the value of the cost function is subject to some uncertain parameters. The target application is the study of complex dynamical systems. It will be shown that when the cost function represents the potential of a dynamical system, the proposed flow‐based model can be used to generate minima conditional to their degree of stability or metastability. In dynamical systems subject to uncertainty in the dynamics, the existence of the minima and their stability characteristics are a function of the uncertain parameters. Thus, the proposed model architecture incorporates a conditional variable that can be the value of the uncertain parameters or a label indicating a characteristic of the critical points. The proposed conditional flow‐model allows the generation of points with the desired characteristics. This is of extreme importance in the analysis of equilibrium states and possible transitions, controlled or uncontrolled, to other equilibrium states. Some illustrative examples of functions with hundreds of local minima are used to test the potentialities of the proposed approach. It will be shown that the use of a generative approach is advantageous to explore more complex landscapes compared to a basic random local search algorithm. When applied to the analysis of the uncertain five body problem, the proposed generative model is shown to successfully identify all dynamical equilibrium solutions under uncertainty. Finally when trained on the dynamical stability properties of the critical points, the model can successfully differentiate between stable and metastable solutions. These results show that, for certain types of system, the flow‐based model can be trained to find equilibrium points more efficiently than a simple random search. Moreover, we demonstrate that conditional flow‐based models are capable of one‐shot sampling for specific values of uncertain parameters or characteristics of the equilibrium points.