Non-variational hysteresis: self-organization and pattern formation
Final Report Abstract
1. Asymptotics of the heat kernels on 2D lattices. Heat kernels, or Green’s functions, are special solutions of reaction-diffusion equations that allow one to analyze general solutions both in the linear and nonlinear cases. One of their important properties is the behavior for large values of spatial or temporal variable. For equations on lattices, the heat kernels have only integral representations, and this behavior can be characterized only in the sense of asymptotics. We analyzed the asymptotics in the case of the two-dimensional spatial variable x. Unlike in the 1D case, it turns out that the asymptotics contains a time independent term Ω(x/ε) which is not rotationally symmetric but rather depends on the polar angle of x. Furthermore, it vanishes for x 6= 0 as ε → 0 and, as such, is an artifact of the spatial discretization. One of the immediate applications of these results would be the analysis of spatio-temporal patterns (rattling) for hysteretic reaction-diffusion equations. 2. Pulses in bistable reaction-diffusion systems with rapidly oscillating coefficients. This topic deals with pulse solutions in FitzHugh–Nagumo systems (coupled parabolic equations) with rapidly periodically oscillating coefficients (depending on x/ε, where x is the spatial variable and ε determines the small period of spatial oscillations). In the limit of vanishing periods, there arises a two-scale FitzHugh–Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We proved existence and stability of pulses in the limit system and showed their proximity to pulse-like solutions of the original system. These pulse-like solutions have a profile with a periodic microstructure. We provided numerical examples for pulses in the original and the two-scale systems. Interestingly, a pulse may exist even if the microscopic average of the inhibitor vanishes at every macroscopic point x. 3. Dynamical systems approach to to robust deep neural networks. This topic deals with learning probability distributions of observed data by artificial neural networks. We introduced a novel approach to reconstructing the ground truth probability distribution based on a gradient conjugate prior (GCP) update. We established a connection between the GCP update and the maximization of the log-likelihood of the predictive distribution. Unlike for the Bayesian neural networks, we used deterministic weights of neural networks, assumed that the ground truth distribution is Gaussian with unknown mean and variance, and learned by the neural networks the parameters of a prior for these unknown mean and variance. We obtained a corresponding dynamical system for the prior’s parameters and analyzed its properties. Next, we analyzed the dynamics and equilibria of the learning process, assuming that the training data set is contaminated by outliers. We showed that the outliers cause a qualitative change in the structure of the energy surfaces of the GCP network. Namely, the global minimum bifurcates from infinity to a finite value. This renders the predictive distribution from Gaussian into Student’s t, whose variance may be significantly larger than the ground truth variance. We showed how the knowledge of the above finite equilibrium allows one to reconstruct the ground truth mean and variance and remove the bias caused by outliers in the training set. Experiments with synthetic and real-world data sets indicate that the GCP network outperforms other state-of-art robust methods.
Publications
- Pulses in FitzHugh Nagumo systems with rapidly oscillating coefficients.
Multiscale Model. Simul., Vol. 16. 2018, Issue 2, pp. 833–856.
Gurevich P., Reichelt S.
(See online at https://doi.org/10.1137/17M1143708) - Spatially discrete reaction-diffusion equations with discontinuous hysteresis. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Vol. 35. 2018, Issue 4, pp. 1041-1077.
Gurevich P., Tikhomirov S.
(See online at https://doi.org/10.1016/j.anihpc.2017.09.006) - Asymptotics of the heat kernels on 2D lattices. Asymptotic Analysis, Vol. 112. 2019, no. 1-2, pp. 107-124.
Gurevich P.
(See online at https://doi.org/10.3233/ASY-181498) - Pairing an arbitrary regressor with an artificial neural network estimating aleatoric uncertainty. Neurocomputing, Vol. 350. 2019, pp. 291-306.
Gurevich P., Stuke H.
(See online at https://doi.org/10.1016/j.neucom.2019.03.031) - Stability of periodic solutions for hysteresis-delay differential
equations. Journal of Dynamics and Differential Equations, Vol. 31. 2019, pp. 1873–1920.
Gurevich P., Ron E.
(See online at https://doi.org/10.1007/s10884-018-9664-0) - Gradient conjugate priors and deep neural networks. Artificial Intelligence, Vol. 278. 2020, 103184.
Gurevich P., Stuke H.
(See online at https://doi.org/10.1016/j.artint.2019.103184)
