Recurrent neural networks with low-rank connectivity matrices are general and tractable models of collective dynamics in large networks. This class of models dates back to the seminal works of J. Hopfield (1982) and S. Amari (1972), and it still plays an instrumental role in computational neuroscience today. To highlight the analytical tractability of these models, I will first review some recent theoretical results concerning the case where the low-rank connectivity is random, the rank is kept fixed, and the number of neurons tends to infinity. In this case, we find that (i) the dynamics of the network converges to a neural field equation, (ii) the dynamics can be reduced to a latent, low-dimensional dynamical system, and (iii) the latent dynamics can be fully solved in certain special cases. In the second part of the presentation, I will show that low-rank connectivity is associated with remarkable concentration of measure phenomena in networks of biological neurons. Considering a feedforward network setup where neurons in the first layer, modelled as Cox processes, transmit stochastic spikes, I will present a theorem stating that the network can behave as if each spiking neuron were transmitting its subthreshold membrane potential as both the rank of the connectivity and the number of neurons tend to infinity. This result could explain how, at the network level, neurons can transmit their subthreshold membrane potential fluctuations through sparse spikes. The proof of the theorem involves the so-called thin shell phenomenon, a well-known concentration phenomenon in high-dimensional probability.

Motivated by recent interest in graphon mean field games and their applications, we study a comprehensive probabilistic analysis of graphon mean field control (GMFC) problems, where the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous mean field interactions. We formulate the GMFC problem with general graphon mean field dependence and establish the existence and uniqueness of the associated graphon mean field forward-backward stochastic differential equations (FBSDEs). We then derive a version of the Pontryagin stochastic maximum principle tailored to GMFC problems. Furthermore, we analyze the solvability of the GMFC problem for linear dynamics and study the continuity and stability of the graphon mean field FBSDEs under the optimal control profile. Finally, we show that the solution to the GMFC problem provides an approximately optimal solution for large systems with heterogeneous mean field interactions, based on a propagation of chaos result. Joint work with Zhongyuan Cao (NYUSH).

In recent years, considerable progress has been made in implementing decision support procedures based on machine learning methods through the use of very large databases and learning algorithms. In many application areas, the available databases are modest in size, raising the question of whether it is reasonable, in this context, to seek to develop powerful tools based on machine learning techniques. This presentation introduces models that leverage various types of knowledge through pre-trained alternative models, targeted observations, or physics in order to implement effective machine learning models in a context of data scarcity. Several industrial applications are used to illustrate benefits of these approaches.

https://www.phnmsbern.ch/public/ph/bilder/Portraits/Mitarbeitende/250801_kirchner_cv.pdf

UMPA, ENS Lyon