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Pavel Gurevich

Publications and source records attributed to Pavel Gurevich.

At least 19 recordsLinked to original sources

HEDGEHOG: Hierarchical Evaluation of Drug Generators Through Rigorous Filtration

Generative molecular models can support early drug discovery by proposing new candidate compounds de novo. In practice, useful candidates must balance target-relevant activity, synthetic accessibility, physicochemical properties, and other multiparameter design constraints. However, metrics commonly used to evaluate molecular generators only weakly reflect whether the generated compounds are medicinally plausible and suitable for downstream computation. This can produce false positives in model evaluation, incorrect assumptions, and inefficient use of computational resources. We introduce HEDGEHOG, a unified six-stage filtration benchmark that is inspired by industrial hit identification workflows: (i) preprocessing; (ii) physicochemical descriptor screening; (iii) structural alerts and graph-sanity checks; (iv) synthesis feasibility; (v) docking and binding affinity estimation; and (vi) three-dimensional pose and interaction checks. We evaluate 23 molecular generators across three model classes under a standardized protocol. Across 230,000 generated molecules, only 0.65% of initial molecules survive all stages. Our results expose a central limitation of current molecular generators: molecules that appear acceptable under isolated criteria rarely satisfy medicinal chemistry, synthesis, docking, and 3D pose filters simultaneously.

cs.LG

Robustness Against Outliers For Deep Neural Networks By Gradient Conjugate Priors

We analyze a new robust method for the reconstruction of probability distributions of observed data in the presence of output outliers. It is based on a so-called gradient conjugate prior (GCP) network which outputs the parameters of a prior. By rigorously studying the dynamics of the GCP learning process, we derive an explicit formula for correcting the obtained variance of the marginal distribution and removing the bias caused by outliers in the training set. Assuming a Gaussian (input-dependent) ground truth distribution contaminated with a proportion $\varepsilon$ of outliers, we show that the fitted mean is in a $c e^{-1/\varepsilon}$-neighborhood of the ground truth mean and the corrected variance is in a $b\varepsilon$-neighborhood of the ground truth variance, whereas the uncorrected variance of the marginal distribution can even be infinite. We explicitly find $b$ as a function of the output of the GCP network, without a priori knowledge of the outliers (possibly input-dependent) distribution. Experiments with synthetic and real-world data sets indicate that the GCP network fitted with a standard optimizer outperforms other robust methods for regression.

stat.ML

Gradient conjugate priors and multi-layer neural networks

The paper deals with learning probability distributions of observed data by artificial neural networks. We suggest a so-called gradient conjugate prior (GCP) update appropriate for neural networks, which is a modification of the classical Bayesian update for conjugate priors. We establish a connection between the gradient conjugate prior update and the maximization of the log-likelihood of the predictive distribution. Unlike for the Bayesian neural networks, we use deterministic weights of neural networks, but rather assume that the ground truth distribution is normal with unknown mean and variance and learn by the neural networks the parameters of a prior (normal-gamma distribution) for these unknown mean and variance. The update of the parameters is done, using the gradient that, at each step, directs towards minimizing the Kullback--Leibler divergence from the prior to the posterior distribution (both being normal-gamma). We obtain a corresponding dynamical system for the prior's parameters and analyze its properties. In particular, we study the limiting behavior of all the prior's parameters and show how it differs from the case of the classical full Bayesian update. The results are validated on synthetic and real world data sets.

math.ST

Asymptotics of the heat kernels on 2D lattices

We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green's functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.

math.DS

Pairing an arbitrary regressor with an artificial neural network estimating aleatoric uncertainty

We suggest a general approach to quantification of different forms of aleatoric uncertainty in regression tasks performed by artificial neural networks. It is based on the simultaneous training of two neural networks with a joint loss function and a specific hyperparameter $λ>0$ that allows for automatically detecting noisy and clean regions in the input space and controlling their {\em relative contribution} to the loss and its gradients. After the model has been trained, one of the networks performs predictions and the other quantifies the uncertainty of these predictions by estimating the locally averaged loss of the first one. Unlike in many classical uncertainty quantification methods, we do not assume any a priori knowledge of the ground truth probability distribution, neither do we, in general, maximize the likelihood of a chosen parametric family of distributions. We analyze the learning process and the influence of clean and noisy regions of the input space on the loss surface, depending on $λ$. In particular, we show that small values of $λ$ increase the relative contribution of clean regions to the loss and its gradients. This explains why choosing small $λ$ allows for better predictions compared with neural networks without uncertainty counterparts and those based on classical likelihood maximization. Finally, we demonstrate that one can naturally form ensembles of pairs of our networks and thus capture both aleatoric and epistemic uncertainty and avoid overfitting.

stat.ML

Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincaré map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev--Slobodeckij spaces and explicitly obtain a formal linearization of the Poincaré map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.

math.DS

Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients

This paper is devoted to pulse solutions in FitzHugh--Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. 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 prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.

math.DS

Spatially discrete reaction-diffusion equations with discontinuous hysteresis

We address the question: Why may reaction-diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called {\it rattling}: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order $a t^{-1/2}$ as $t\to\infty$ and explicitly find the rate $a$.

math.AP

Asymptotics of parabolic Green's functions on lattices

For parabolic spatially discrete equations, we consider Green's functions, also known as heat kernels on lattices. We obtain their asymptotic expansions with respect to powers of time variable $t$ up to an arbitrary order and estimate the remainders uniformly on the whole lattice. The spatially discrete (difference) operators under consideration are finite-difference approximations of continuous strongly elliptic differential operators (with constant coefficients) of arbitrary even order in $\mathbb R^d$ with arbitrary $d\in\mathbb N$. This genericity, besides numerical and deterministic lattice-dynamics applications, allows one to obtain higher-order asymptotics of transition probability functions for continuous-time random walks on $\mathbb Z^d$ and other lattices.

math.AP

Dynamics of Discrete Time Systems with a Hysteresis Stop Operator

We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by modifications to general-equilibrium macroeconomic models that attempt to capture the frictions and memory-dependence of realistic economic agents.

math.DS

Rattling in spatially discrete diffusion equations with hysteresis

The paper treats a reaction-diffusion equation with hysteretic nonlinearity on a one-dimensional lattice. It arises as a result of the spatial discretization of the corresponding continuous model with so-called nontransverse initial data and exhibits a propagating microstructure --- which we call {\em rattling} --- in the hysteretic component of the solution. We analyze this microstructure and determine the speed of its propagation depending on the parameters of hysteresis and the nontransversality coefficient in the initial data.

math.AP

Asymptotics of sign-changing patterns in hysteretic systems with diffusive thresholds

We consider a reaction-diffusion system including discontinuous hysteretic relay operators in reaction terms. This system is motivated by an epigenetic model that describes the evolution of a population of organisms which can switch their phenotype in response to changes of the state of the environment. The model exhibits formation of patterns in the space of distributions of the phenotypes over the range of admissible switching strategies. We propose asymptotic formulas for the pattern and the process of its formation.

math.AP

Recent Advances in Reaction-Diffusion Equations with Non-Ideal Relays

We survey recent results on reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time, and also describe its regularity. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism --- {\it rattling}, which indicates how one should reset the continuous model to make it well posed.

math.AP

Stability for semilinear parabolic problems in $L_2$, $W^{1,2}$, and interpolation spaces

An asymptotic stability result for parabolic semilinear problems in $L_2(Ω)$ and interpolation spaces is shown. Some known results about stability in $W^{1,2}(Ω)$ are improved for semilinear parabolic mixed boundary value problems. The approach is based on Amann's power extrapolation scales. In a Hilbert space setting, a better understanding of this approach is provided for operators satisfying Kato's square root problem; as a side result some equivalent characterizations of these operators are obtained.

math.AP

Pattern formation in parabolic equations containing hysteresis with diffusive thresholds

We consider a reaction-diffusion system with discontinuous reaction terms modeled by non-ideal relays. The system is motivated by an epigenetic population model of evolution of two-phenotype bacteria which switch phenotype in response to variations of environment. We prove the formation of patterns in the phenotype space. The mechanism responsible for pattern formation is based on memory (hysteresis) of the non-ideal relays.

math.AP

Well-posedness of parabolic equations containing hysteresis with diffusive thresholds

We study complex systems arising, in particular, in population dynamics, developmental biology, and bacterial metabolic processes, in which each individual element obeys a relatively simple hysteresis law (a non-ideal relay). Assuming that hysteresis thresholds fluctuate, we consider the arising reaction-diffusion system. In this case, the spatial variable corresponds to the hysteresis threshold. We describe the collective behavior of such a system in terms of the Preisach operator with time-dependent measure which is a part of the solution for the whole system. We prove the well-posedness of the system and discuss the long-term behavior of solutions.

math.AP