F1 Прикладна математика

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https://ekmair.ukma.edu.ua/handle/123456789/18299

Освітня програма: "Прикладна математика"

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Active reinforcement learning for data-efficient digital twin calibration
(2026) Спiтковська, Владислава ; Кузьменко, Дмитро
Digital twin calibration often faces a critical problem: classical methods treat data collection and parameter fitting as independent stages, whereas informative observations may need to be actively generated under budget constraints. This makes purely passive approaches non-adaptive and poorly scalable in online calibration tasks. In this work, active digital twin calibration is formulated as a partially observable Markov decision process with hidden environment parameters and an explicit query budget. A unified framework is proposed that jointly optimises trajectory generation, parameter estimation, and query timing. The proposed pipeline consists of three components: a PPO-based controller that generates informationally rich trajectories through reward shaping, a GRUbased recurrent identifier that produces both a point estimate ^𝜃(𝑡) and an uncertainty signal 𝜎(𝑡), trained under a Gaussian negative log-likelihood loss, and a PPO calibration policy that makes binary oracle query decisions under a hard budget constraint. The main focus of this work is on the first two parts of the pipeline. The framework is evaluated on two environments of increasing complexity. In the Pendulum environment with a single parameter 𝜃, the GRU-only mode achieves a 96% error reduction compared to passive baselines without any oracle queries. The calibrator operating directly with the GRU achieves MAE(𝑔) = 0.00917 while staying within the predefined query budget. The second environment, WaterWorld, has three hidden parameters. The final GRU approximator achieves a mean normalised error of approximately 5% across all policies. Empirical studies show that a controller oriented toward environment exploration achieves approximately one-quarter lower identification error compared to a task-optimal but informationally passive controller. This supports the conclusion that policy selection and data quality are directly interdependent in active calibration settings.
Active Viewpoint Planning for Efficient 3D Reconstruction
(2026) Кононович, Софiя ; Кузьменко, Дмитро
Efficient data acquisition is a key factor in neural 3D reconstruction, where redundant viewpoints can increase computational cost without improving reconstruction quality. We propose a reinforcement learning-based active viewpoint policy that optimises image collection for NeRF reconstruction under a fixed sensing budget. The agent is trained to maximise scene coverage while penalising redundant observations and inefficient trajectories, and learns to terminate data collection as information gain saturates. Evaluated on indoor scenes from the Replica dataset, our method achieves 33.33 dB PSNR while reducing the number of views by 25% (from 80 to 60), with a proportional reduction in scene acquisition time. Under the same sensing budget, it outperforms a coverage-only policy (33.05 dB) and a heuristic baseline (18.79 dB). The learned policy generalises to unseen scenes without retraining. These results indicate that learned data acquisition can improve both efficiency and reconstruction quality as an upstream component of 3D pipelines.
Binary relations between binary operations
(2021) Bilyi, Illia; Козеренко, Сергій
Let Bin(X) be a collection of all groupoids on some non-empty set X. De ne the operation : Bin2(X) ! Bin(X) so that x( )y = (x y) (y x) for all x; y 2 X and (X; ); (X; ) 2 Bin(X). Let lz denote left-zero operation (8x; y 2 X : x lz y = x) on X. Then, (X; lz) is an identity of (Bin(X); ). Similarly, de ne right-zero rz 2 Bin(X) (8x; y 2 X : x rz y = y). We consider the center of (Bin(X); ) and represent its elements as graphs. Furthermore, we investigate distributivity from the left in Bin(X) and its interaction with -product. We show that the only operation that is left- distributive over all possible 2 Bin(X) is rz 2 Bin(X) and that any 2 Bin(X) is left-distributive over lz; rz 2 Bin(X).
Boolean network optimization by stochastic rewiring
(2024) Черевко, Крiстiна; Олійник, Богдана
This bachelor thesis introduces novel algorithms for the area minimization of Multi-Input And-Inverter Graphs (MAIGs) within the field of logic synthesis. The primary focus is on the implementation of a new method named "Boolean network optimization by stochastic rewiring". This approach significantly diverges from traditional techniques like algebraic factoring and AIG rewriting by adopting a global strategy to add and remove wires at various locations within the circuit. The method relies on a simple randomization strategy to produce structural variations in the AIGs. Additionally, the thesis formulates and proves a new criterion for acceptable fanins, which ensures valid insertions and deletions without compromising the functionality of the circuit. Experimental results validate the effectiveness of the proposed algorithm in optimizing benchmark circuits. This work is important as it addresses the growing demand for smaller, more efficient circuits in modern electronics, potentially reducing manufacturing costs and enhancing performance.
Coherence in the coupled oscillators for the case of financial time series
(2020) Марченко, Анастасія; Щестюк, Наталія
The analysis in natural science leads to spreading the ideas of chaos theory and non- linear dynamics to nancial mathematics and creating the new researches to consider similar models and procedures for nancial time series. Also, the irregular uctuations in these series are sometimes considered as an outcome from chaotic systems.[1] This can be used, for example, to forecast the value of an investment portfolio, which is the combination of di erent nancial assets, for example, stocks, bonds, cash. One of the ways to think about a successful portfolio is when the chosen equities have the high expected returns and synchronized in time for bottom moments.[2] Then the dynamics of these nancial assets can be described as oscillators connected in the network.
Combining machine learning with physical laws to solve inverse problems
(2026) Надєєв, Iлля ; Крюкова, Галина
This bachelor’s thesis investigates the use of neural-network-based methods for solving forward and inverse problems in dynamical systems. The work considers neural networks (NNs) and Physics-Informed Neural Networks (PINNs). The thesis provides a description of differential equations, inverse problems, and the theoretical foundations of Physics-Informed Neural Networks. The practical part considers Euler’s planar two-fixed-centers problem as a model dynamical system for studying forward and inverse PINN formulations. Numerical experiments are conducted for both forward trajectory reconstruction and inverse parameter recovery.
Developing a Hybrid AI model for Financial Market Prediction
(2025) Войтішин, Микита; Кузьменко, Дмитро
Over the recent years, financial time series modeling has presented a significant challenge due to market stochasticity and volatility. The stock market is influenced not only by market data such as price and volume but also by a wide range of additional external factors, including macroeconomic indicators, seasonality, fundamentals, and market sentiment. The increasing availability of diverse financial data, combined with the rapid advances in artificial intelligence (AI), has opened up new possibilities for analyzing and understanding how stock markets behave. These technologies have the potential to capture more complex nonlinear patterns that traditional statistical and machine learning models often fail to detect. This research examines how combining various model architectures and feature sets with domain - specific knowledge from the financial sector can enhance uncertainty quantification, a crucial aspect of making informed decisions and investments in financial markets.
Development of a Control Algorithm for a Two-Axis Gimbal for Object Tracking on UAV
(2026) Kulyk, Zorian ; Kurochkin, Andrew
This work presents the design, implementation, and evaluation of a control algorithm for active object tracking on a low-cost, two-axis (pitch-roll) brushless gimbal intended for UAV deployment. The platform combines a Raspberry Pi 4 for vision processing, a SpeedyBee F405 flight controller communicating over MAVLink, an IMX378 camera, and a gimbal based on an 8-bit controller board. The vision pipeline is built entirely on OpenCV and compares two correlation-filter trackers, MOSSE and CSRT. Pixel error from the tracker is mapped directly to gimbalPWMcommands through a proposed conversion with a spatial resolution coefficient derived from the camera intrinsics. The work also derives the forward kinematics of the two-axis pitch–roll configuration and establishes its theoretical limitations. Experimental results obtained on the assembled static platform provide an evaluation of the object tracking error and confirm that the proposed approach is feasible within the stated hardware constraints.
Development of a system for physical exercises analysis and fitness facilities optimization using computer vision and machine learning
(2026) Тиндик, Маркіян ; Кузьменко, Дмитро
This thesis builds a small computer vision system that watches a person doing exercises and feeds what it sees into a layout optimizer for the gym they are using. The whole thing was put together with cheap hardware: two USB webcams, two IP cameras, a steel tape, and a smartphone level app, in a single room of about 33 m2. The pipeline runs on two paths. Offline, it uses MediaPipe PoseLandmarker on CPU to extract 33 body landmarks per frame, computes joint angles, and counts repetitions by finding valleys in the smoothed angle signal. Live, it uses YOLO11n-pose on CUDA, adds ArcFace face recognition with a hysteresis layer for stable identities, runs phase-gated rep counters in the style of fitness apps like Fitton, and fuses up to four cameras into a single 2D room map. A MILP places ten machine types on a 5×4 grid, and a heatmap built from live position logs re-weights the cost matrix and re-solves it. The maths is derived in the methods chapter: pinhole projection, ray-to-floor intersection, the dot-product joint-angle identity, weighted least-squares fusion, and the linear and quadratic assignment formulations. Headline numbers across six experiments: 99.1 / 96.7 / 93.0% pose detection at 72 to 80 FPS on a single CPU core; 6 / 3 / 3 reps on the squat / push-up / pull-up clips; 35.64% MILP cost reduction over a random layout; 43.42% extra improvement once the heatmap is plugged in; 90% zone-classification accuracy on a controlled fourzone walk.
Dropout for Neural Networks Pruning
(2025) Семенець, Дарина; Швай, Надія
In this study, the hypothesis is examined whether Dropout masks can be used for structural pruning without further evaluating the importance of individual filters or weight. It was decided to compare a Dropout-based approach, which is based on the utilization of binary Dropout masks, with a classical L2-Norm-based pruning method. For this task, we manually designed an architecture of a convolutional neural network with a custom Dropout. The research undergoes the following phases: designing a mask generation mechanism, preprocessing data, training of model, implementing of pruning algorithms, and conducting experiments using the Imagenette2 dataset. Our idea is to determine whether Dropout pruning can offer a reliable alternative to traditional methods, especially under different levels of sparsity and stochasticity.
Enhancing Temporal Smoothing in Dynamic Neural Radiance Fields
(2025) Вербицька, Марія; Кузьменко, Дмитро
In this work, we conduct an end-to-end training and fine-tuning process for the Neural Radiance Field (NeRF) model [1] and introduce 4 experimental cases with filtering techniques [2] designed to strengthen the rendering performance. We evaluate our modifications on synthetic image data of the articulated objects. For this project, we chose the architecture of the Knowledge NeRF model [3]. It includes an original PyTorch NeRF implementation [4] alongside a projection module for dynamic scenes extension. Incorporating the rendering step adjustments allows for better results without requiring complete model re-training. Our study covers the theoretical basis of the 3D scene reconstruction problem [5] alongside the NeRF architecture, such as radiance field, volume rendering, the concept of coarse and fine networks etc. [1], provides a trained and fine-tuned model for one object of a specified motion type, and suggests four methods to handle postprocessing in Knowledge NeRF better.
Estimating the probability of rare events using the Monte Carlo method: the example of the duration of a ceasefire
(2026) Столяр, Яна ; Мацукi, Йошiо
The qualification thesis is devoted to the study of estimating the probabilities of rare events using the Monte Carlo method, with the duration of ceasefires as a case study. The main objective is to develop and validate a simulationbased approach for estimating the probabilities of events that occur rarely, and to compare the obtained results with analytical models based on the exponential distribution. In this work, the Monte Carlo method with repeated random sampling is implemented. The convergence of the obtained estimates is analyzed, and the limitations of classical analytical methods in modeling extreme events are investigated. The study demonstrates an integrative approach that combines theoretical probability with computational experimentation and highlights the practical value of simulation methods for verifying analytical assumptions.
Hybrid LSTM-GARCH Architecture for Option Pricing
(2026) Pasternak, Olena ; Shchestyuk, Nataliia
The bachelor’s thesis compares the accuracy of a proposed dual-branch Long Short-TermMemory (LSTM) neural network, processing 14-day historical sequences and GARCH-derived volatility, with the classical Black-Scholes-Merton model for pricing European call options. Empirical testing on approximately 800,000 Apple Inc. (AAPL) option records demonstrates that the LSTM network efficiently captures complex non-linear dependencies and market dynamics. At the first forecast step, the model achieved a 63.8% reduction in Mean Absolute Error (MAE) compared to the BSM-GARCH variant, while maintaining high predictive stability over a five-day horizon.
Image embeddings with Kolmogorov-Arnold networks
(2025) Юрченко, Артур; Кузьменко, Дмитро
This research aims to evaluate performance of Kolmogorov-Arnold networks (KAN) in image embedding tasks. It focuses on modifying existing state-of-the-art architectures - CNN and ViT, replacing their MLP segments with KANs, aiming to improve their computational performance and embedding quality. Training and evaluation methodology is fully described in sections 4 and 5.
Left-distributivity relation on the semigroup Bin(X)
(2022) Krolevets, Mariia; Kozerenko, Serhiy
Let X be a nonempty set. Bin(X) is the collection of all groupoids defined on X. Let ; 2 Bin(X). We define a binary operation on Bin(X) as follows: 8x; y 2 X : x[ ]y = (x y) (y x): In fact, (Bin(X); ) is a monoid with left-zero operation lz being its identity, where 8x; y 2 X : x lz y = x. ZBin(X) is the set of all elemets of Bin(X) that commute with every other elements under . In this thesis, we study the left-distributivity relation on the semigroup Bin(X) and the group ZBin(X).We research the question of trivial left-distributivity neighborhoods in Bin(X). Furthermore, we give a criterion, which characterizes those elements of ZBin(X), which the given element distributes from the left with.
LLM-based Smart Chatbot Design Using LangChain Framework
(2024) Колодяжна, Дарія; Кузьменко, Дмитро
This work presents the development of a smart chatbot called "Legislative Assistant" that assists Ukrainian lawyers and citizens in navigating legislative acts. The chatbot utilizes large language models (LLMs), specifically the Mistral-7B-Instruct-Ukrainian model, integrated with the LangChain framework and retrieval-augmented generation (RAG) techniques. By creating embeddings from a corpus of Ukrainian legislative acts and optimizing retrieval using RAG, the chatbot provides accurate and up-to-date responses to queries related to legislation. This work covers theoretical aspects such as the evolution of language models, transformer architectures like Mistral 7B and the concept of retrieval-augmented generation (RAG) for incorporating external data into language model responses.
M-Лiпшицевi вiдображення на графах
(2024) Гуназа, Анна; Козеренко, Сергій
Квалiфiкацiйна робота присвячена дослiдженню M-Лiпшицевих вiдображень на графах. У роботi розроблено та проаналiзовано алгоритми для задач M-ParExt та Strong M-ParExt на деревах, загальних графах та графах блокiв. Вони дозволяють розширювати частковi M-Лiпшицевi вiдображення на весь граф, зберiгаючи при цьому властивiсть M-досяжностi мiж образами вершин. Представлено теоретичне обґрунтування, програмну реалiзацiю на Python та приклади застосування.
Modelling prosody in the task of human speech synthesis with the use of machine learning
(2020) Процик, Олексій; Крюкова, Галина
Generating high fidelity speech using a text-to-speech (TTS) system remains a challenging task despite the decades of research and investigations. Modern TTS systems are very complex. For example, it is a common practice for a statistical TTS system to have a linguistic extractor in the front, which extracts different linguistic features. It is followed by a duration model to estimate the speech length in time of a given text and an acoustic feature prediction model. Given these features, it is all fed into a vocoder, which synthesizes speech out of acoustic features. All these components are trained independently and require extensive field knowledge to be sophisticated enough and produce considerable results. Because it has a modular design, it is prone to errors which will proceed in the following modules and can accumulate.
NK-моделі Кауффмана
(2020) Крошин, Олександр; Чорней, Руслан
A given work focuses on the NK fitness models, focuses on classical NK models of Boolean networks dynamics, gives intuition of its basic properties and describes most useful ways to represent it. The paper is also focused on a fitness landscape and its corresponding NK model, concepts of ruggedness and smoothness. It also covers a concept of NK-models stability and researches an impact of internal and external parameters of a model (N, K, ruggedness of a landscape) on its stability. Experiments were implemented with Python 3.8 using libraries Numpy, matplotlib etc.
Non linear stochastic models for time series analysis of stock volatilities
(2022) Fisun, Yelyzaveta; Shchestyuk, Natalia
Квалiфiкацiйну роботу присвячено застосуванню теоретичних основ нелiнiйних стохастичних моделей, а саме ARCH(p), GARCH(p,q) на реальних фiнансових даних. У роботi проведена оцiнка методом моментiв та методом максимальної вiрогiдностi, їх порiвняння та симуляцiя моделей. Спрогнозована поведiнка волатильностi акцiй на певний перiод.

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