arXiv subjects

Alexander Tarakanov

Publications and source records attributed to Alexander Tarakanov.

8 recordsLinked to original sources

arXiv2026

PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=\Lambda(x)U(x)$. The diagonal factor $\Lambda(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

cs.LG↗

arXiv2026

Loss-Conditional PINNs for Parametric PDE Families

Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses. Their performance is often dominated by the choice of loss weights: a poor weighting can drive training to a degenerate solution in which one physical constraint is satisfied while another is ignored. Existing methods select or adapt a single good set of weights. We take a different view: instead of tuning one weight vector, we explore the entire weight space during training. We introduce LC-PINN, which adapts the loss-conditional training of Dosovitskiy and Djolonga (2020) to the PDE-residual setting: the conditioning vector (either the loss weights or a scalar physical coefficient) is treated as a network input and sampled from a simple prior at every optimisation step. This turns PINN training into learning a continuous family of solutions indexed by that vector, with no solver-generated paired data. LC-PINN thus lies between classical PINNs and operator learning: it stays fully physics-informed but amortises training over a parametric family. Our contribution is not the loss-conditional construction itself, but its extension to PINNs, the unification of the loss-weight and parametric-coefficient regimes under one architecture (concatenation for loss weights, FiLM for coefficients), and a fixed-quadrature L-BFGS finishing protocol that makes the parametric-coefficient regime trainable. We give a lambda-invariance result for the conditional optimum and study LC-PINN on parametric Helmholtz, Schrodinger, viscous Burgers, and Buckley-Leverett equations. A single LC-PINN matches or improves retrained per-weight PINN baselines while parameterising the full family in one model, at a total cost that amortises favourably against per-instance retraining.

cs.LG↗

arXiv2024

Digital Fingerprinting of Microstructures

Finding efficient means of fingerprinting microstructural information is a critical step towards harnessing data-centric machine learning approaches. A statistical framework is systematically developed for compressed characterisation of a population of images, which includes some classical computer vision methods as special cases. The focus is on materials microstructure. The ultimate purpose is to rapidly fingerprint sample images in the context of various high-throughput design/make/test scenarios. This includes, but is not limited to, quantification of the disparity between microstructures for quality control, classifying microstructures, predicting materials properties from image data and identifying potential processing routes to engineer new materials with specific properties. Here, we consider microstructure classification and utilise the resulting features over a range of related machine learning tasks, namely supervised, semi-supervised, and unsupervised learning. The approach is applied to two distinct datasets to illustrate various aspects and some recommendations are made based on the findings. In particular, methods that leverage transfer learning with convolutional neural networks (CNNs), pretrained on the ImageNet dataset, are generally shown to outperform other methods. Additionally, dimensionality reduction of these CNN-based fingerprints is shown to have negligible impact on classification accuracy for the supervised learning approaches considered. In situations where there is a large dataset with only a handful of images labelled, graph-based label propagation to unlabelled data is shown to be favourable over discarding unlabelled data and performing supervised learning. In particular, label propagation by Poisson learning is shown to be highly effective at low label rates.

cs.CV↗

arXiv2021

Randomized multilevel Monte Carlo for embarrassingly parallel inference

This position paper summarizes a recently developed research program focused on inference in the context of data centric science and engineering applications, and forecasts its trajectory forward over the next decade. Often one endeavours in this context to learn complex systems in order to make more informed predictions and high stakes decisions under uncertainty. Some key challenges which must be met in this context are robustness, generalizability, and interpretability. The Bayesian framework addresses these three challenges elegantly, while bringing with it a fourth, undesirable feature: it is typically far more expensive than its deterministic counterparts. In the 21st century, and increasingly over the past decade, a growing number of methods have emerged which allow one to leverage cheap low-fidelity models in order to precondition algorithms for performing inference with more expensive models and make Bayesian inference tractable in the context of high-dimensional and expensive models. Notable examples are multilevel Monte Carlo (MLMC), multi-index Monte Carlo (MIMC), and their randomized counterparts (rMLMC), which are able to provably achieve a dimension-independent (including $\infty-$dimension) canonical complexity rate with respect to mean squared error (MSE) of $1/$MSE. Some parallelizability is typically lost in an inference context, but recently this has been largely recovered via novel double randomization approaches. Such an approach delivers i.i.d. samples of quantities of interest which are unbiased with respect to the infinite resolution target distribution. Over the coming decade, this family of algorithms has the potential to transform data centric science and engineering, as well as classical machine learning applications such as deep learning, by scaling up and scaling out fully Bayesian inference.

stat.CO↗

arXiv2021

On Unbiased Estimation for Discretized Models

In this article, we consider computing expectations w.r.t. probability measures which are subject to discretization error. Examples include partially observed diffusion processes or inverse problems, where one may have to discretize time and/or space, in order to practically work with the probability of interest. Given access only to these discretizations, we consider the construction of unbiased Monte Carlo estimators of expectations w.r.t. such target probability distributions. It is shown how to obtain such estimators using a novel adaptation of randomization schemes and Markov simulation methods. Under appropriate assumptions, these estimators possess finite variance and finite expected cost. There are two important consequences of this approach: (i) unbiased inference is achieved at the canonical complexity rate, and (ii) the resulting estimators can be generated independently, thereby allowing strong scaling to arbitrarily many parallel processors. Several algorithms are presented, and applied to some examples of Bayesian inference problems, with both simulated and real observed data.

stat.CO↗

arXiv2020

Optimal Bayesian experimental design for subsurface flow problems

Optimal Bayesian design techniques provide an estimate for the best parameters of an experiment in order to maximize the value of measurements prior to the actual collection of data. In other words, these techniques explore the space of possible observations and determine an experimental setup that produces maximum information about the system parameters on average. Generally, optimal Bayesian design formulations result in multiple high-dimensional integrals that are difficult to evaluate without incurring significant computational costs as each integration point corresponds to solving a coupled system of partial differential equations. In the present work, we propose a novel approach for development of polynomial chaos expansion (PCE) surrogate model for the design utility function. In particular, we demonstrate how the orthogonality of PCE basis polynomials can be utilized in order to replace the expensive integration over the space of possible observations by direct construction of PCE approximation for the expected information gain. This novel technique enables the derivation of a reasonable quality response surface for the targeted objective function with a computational budget comparable to several single-point evaluations. Therefore, the proposed technique reduces dramatically the overall cost of optimal Bayesian experimental design. We evaluate this alternative formulation utilizing PCE on few numerical test cases with various levels of complexity to illustrate the computational advantages of the proposed approach.

physics.comp-ph↗

arXiv2019

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Surrogate-modelling techniques including Polynomial Chaos Expansion (PCE) is commonly used for statistical estimation (aka. Uncertainty Quantification) of quantities of interests obtained from expensive computational models. PCE is a data-driven regression-based technique that relies on spectral polynomials as basis-functions. In this technique, the outputs of few numerical simulations are used to estimate the PCE coefficients within a regression framework combined with regularization techniques where the regularization parameters are estimated using standard cross-validation as applied in supervised machine learning methods. In the present work, we introduce an efficient method for estimating the PCE coefficients combining Elastic Net regularization with a data-driven feature ranking approach. Our goal is to increase the probability of identifying the most significant PCE components by assigning each of the PCE coefficients a numerical value reflecting the magnitude of the coefficient and its stability with respect to perturbations in the input data. In our evaluations, the proposed approach has shown high convergence rate for high-dimensional problems, where standard feature ranking might be challenging due to the curse of dimensionality. The presented method is implemented within a standard machine learning library (Scikit-learn) allowing for easy experimentation with various solvers and regularization techniques (e.g. Tikhonov, LASSO, LARS, Elastic Net) and enabling automatic cross-validation techniques using a widely used and well tested implementation. We present a set of numerical tests on standard analytical functions, a two-phase subsurface flow model and a simulation dataset for CO2 sequestration in a saline aquifer. For all test cases, the proposed approach resulted in a significant increase in PCE convergence rates.

cs.CE↗

arXiv2012

On some discrete subgroups of the Lorentz group

Some discrete subgroups of the Lorentz group are found using Fedorov's parametrization by means of complex vector-parameter. It is shown that the discrete subgroup of the Lorentz group, which have not fixed points, are contained in boosts along a spatial direction for time-like and space-like vectors and are discrete subgroups of the group SO(1,1), whereas discrete subgroups of isotropic vector are subgroups of SO(1,1)\times E(1,1).

physics.gen-ph↗