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

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.