Neural Chains and Discrete Dynamical Systems

We inspect the analogy between machine-learning (ML) applications based on the transformer architecture without self-attention, {\it neural chains} hereafter, and discrete dynamical systems associated with discretised versions of neural integral and partial differential equations (NIE, PDE). A comparative analysis of the numerical solution of the (viscid and inviscid) Burgers and Eikonal equations via standard numerical discretization (also cast in terms of neural chains) and via PINN's learning is presented and commented on. It is found that standard numerical discretization and PINN learning provide two different paths to acquire essentially the same knowledge about the dynamics of the system. PINN learning proceeds through random matrices which bear no direct relation to the highly structured matrices associated with finite-difference (FD) procedures. Random matrices leading to acceptable solutions are far more numerous than the unique tridiagonal form in matrix space, which explains why the PINN search typically lands on the random ensemble. The price is a much larger number of parameters, causing lack of physical transparency (explainability) as well as large training costs with no counterpart in the FD procedure. However, our results refer to one-dimensional dynamic problems, hence they don't rule out the possibility that PINNs and ML in general, may offer better strategies for high-dimensional problems.

A kinetic-based regularization method for data science applications

We propose a physics-based regularization technique for function learning, inspired by statistical mechanics. By drawing an analogy between optimizing the parameters of an interpolator and minimizing the energy of a system, we introduce corrections that impose constraints on the lower-order moments of the data distribution. This minimizes the discrepancy between the discrete and continuum representations of the data, in turn allowing to access more favorable energy landscapes, thus improving the accuracy of the interpolator. Our approach improves performance in both interpolation and regression tasks, even in high-dimensional spaces. Unlike traditional methods, it does not require empirical parameter tuning, making it particularly effective for handling noisy data. We also show that thanks to its local nature, the method offers computational and memory efficiency advantages over Radial Basis Function interpolators, especially for large datasets.