The optimal mass transport problem is a classical problem in mathematics, and dates back to 1781 and work by G. Monge where he formulated an optimization problem for minimizing the cost of transporting soil for construction of forts and roads. Historically the optimal mass transport problem has been widely used in economics in, e.g., planning and logistics, and was at the heart of the 1975 Nobel Memorial Prize in Economic Sciences. In the last two decades there has been a rapid development of theory and methods for optimal mass transport and the ideas have attracted considerable attention in several economic and engineering fields. These developments have led to a mature framework for optimal mass transport with computationally efficient algorithms that can be used to address problems in the many areas.

In this talk, we will consider optimization problems consisting of optimal transport costs together with other functionals to address inverse problems in many domains, e.g., in medical imaging, radar imaging, and spectral estimation. This is a flexible framework and allows for incorporating forward models, specifying dynamics of the object and other dependencies. These problem can often be formulated as a multi-marginal optimal transport problem and we show how common problems, such as barycenter and tracking problems, can be seen as special cases of this. This naturally leads to consider structured optimal transport problems, which can be solved efficiently using customized methods inspired by the Sinkhorn iterations.

PDE GCNNs are neural networks where each layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer’s trainable weights. In this talk, we present a contribution on building new layers that are based either on Wasserstein gradient flows or on normalizing measures that take inspiration from optimal transport maps. The tunable parameters are either connected to parameters of the gradient flow, or the transport maps, so the whole procedure can be interpreted as an inverse problem.

Diffusion Magnetic Resonance Imaging (dMRI) is a non-invasive imaging technique that draws structural information from the interaction between water molecules and biological tissues. Common ways of tackling the derived inverse problem include, among others, Diffusion Tensor Imaging (DTI), High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Spectrum Imaging (DSI). However, these methods are structurally unable to recover the full diffusion distribution, only providing partial information about particle displacement. In our work, we introduce a Total-Variation (TV) regularization defined from an optimal transport perspective using 1-Wasserstein distances. Such a formulation produces a variational problem that can be handled by well-known algorithms enjoying good convergence properties, such as the primal-dual proximal method by Chambolle and Pock. It allows for the reconstruction of the complete diffusion spectrum from measured undersampled k/q space data.

The modelling of the dynamic of a system of rational agents may take inspiration from various sources, depending on the particular application one has in mind. We can consider for example to model interactions via a Newtonian-like system, taking inspiration from physics, or via a game-based approach stemming from classical game theory or mean field games. In both cases, once we ensured the well-posedness of the proposed model, the model itself can be used as a tool to learn from real world observations, by means of learning (some) unknown components of it.

In [1], the authors study a class of spatially inhomogeneous evolutionary games to model the interactions between a finite number of agents: each agent evolves in space with a velocity which depends on a certain underlying mixed strategy, in turn evolving according to a replicator dynamic. In this talk we move from such a formulation, and introduce an entropic limiting version of it, which boils down to a purely spatial ODE. For a bounded set of pure strategies $U \subset \mathbb{R}^u$, $0 < \eta \in P(U)$ a probability measure on $U$, an ''entropic'' parameter $\varepsilon>0$, and maps $e \colon \mathbb{R}^d \times U \to \mathbb{R}$ and $J \colon \mathbb{R}^d \times U \times \mathbb{R}^d \to \mathbb{R}$, the $N$-agents system we consider is the following: $$ \begin{aligned} \partial_t x_i(t) &= v_i^J(x_1(t),\dots,x_N(t)) \quad \text{for } i = 1,\dots,N\\ v_i^J(x_1,\dots,x_N) &= \int_U e(x_i,u)\, \sigma_{i}^J(x_1,\ldots,x_N)(u)\,\,{d} \eta(u)\\ \sigma_{i}^J(x_1,\ldots,x_N) &= \frac{\exp\left(\tfrac{1}{\varepsilon N}\sum_{j=1}^N J(x_i,\cdot,x_j)\right)}{ \int_U \exp\left(\tfrac{1}{\varepsilon N}\sum_{j=1}^N J(x_i,v,x_j)\right)\,\,{d} \eta(v)}. \\ \end{aligned} $$ We study the well-posedness and the mean field limit of such a system, and use it as the backbone of a learning procedure. In particular, we focus on the learnability of the interaction kernel $J$, all the rest given. Building on ideas of [3, 4, 5], we infer $J$ by penalizing the empirical mean squared error between observed velocities and predicted velocities, and also consider the choice of penalizing observed mixed strategies and predicted mixed strategies. We study the quality of the inferred kernel both as $N$ increases (i.e., as we have observations of an increasingly high number of agents) and in the limit of repeated observations with fixed $N$ (i.e., as we have repeated observations of the same number of agents). We show the effectiveness of the proposed inference on many different examples, from classical Newtonian systems to system modelling pedestrian dynamics.

[1] L. Ambrosio, M. Fornasier, M. Morandotti, G. Savaré. Spatially inhomogeneous evolutionary games, Communications on Pure and Applied Mathematics 74.7: 1353-1402, 2021.

[2] M. Bonafini, M. Fornasier, B. Schmitzer. Data-driven entropic spatially inhomogeneous evolutionary games, European Journal of Applied Mathematics 34.1: 106-159, 2023.

[3] M. Bongini, M. Fornasier, M. Hansen, M. Maggioni. Inferring interaction rules from observations of evolutive systems I: The variational approach, Mathematical Models and Methods in Applied Sciences 27.05: 909-951, 2016.

[4] F. Cucker, S. Smale. On the mathematical foundations of learning, Bulletin of the American mathematical society 39.1: 1-49, 2002.

[5] F. Lu, M. Maggioni, S. Tang. Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, The Journal of Machine Learning Research 22.1: 1518-1584, 2021.