arXiv subjects

Domagoj Vlah

Publications and source records attributed to Domagoj Vlah.

14 recordsLinked to original sources

arXiv2026

Quartic Rational Diophantine Quadruples and the Euler Surface

We prove that there exist infinitely many quartic rational Diophantine quadruples, that is, sets of four pairwise distinct nonzero rational numbers whose pairwise products increased by 1 are fourth powers in Q. To the best of our knowledge, no examples of such quadruples were previously known. Our construction is motivated by computer experiments and leads naturally to the classical Euler surface E:X^4+Y^4=Z^4+W^4. We show that every rational point on a suitable Zariski-open subset of E yields a quartic rational Diophantine quadruple, thereby obtaining a rational map from the Euler surface to the parameter space of quartic quadruples. In particular, Euler's classical parametrization produces the first explicit infinite family of quartic rational Diophantine quadruples. We also explain that the same mechanism extends to arbitrary exponents k>1, with the Euler surface replaced by the Fermat--Euler surface E_k:X^k+Y^k=Z^k+W^k. For even k, every rational point on a suitable open subset of E_k gives rise to a kth power rational Diophantine quadruple, while for odd k one obtains such quadruples on the locus where W/Z is a square.

math.NT↗

arXiv2026

AE-ViT: Stable Long-Horizon Parametric Partial Differential Equations Modeling

Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear mappings, and utilize GPUs. Existing approaches typically learn evolution either on the full solution field, which requires capturing long-range spatial interactions at high computational cost, or on compressed latent representations obtained from autoencoders, which reduces the cost but often yields latent vectors that are difficult to evolve, since they primarily encode spatial information. Moreover, in parametric PDEs, the initial condition alone is not sufficient to determine the trajectory, and most current approaches are not evaluated on jointly predicting multiple solution components with differing magnitudes and parameter sensitivities. To address these challenges, we propose a joint model consisting of a convolutional encoder, a transformer operating on latent representations, and a decoder for reconstruction. The main novelties are joint training with multi-stage parameter injection and coordinate channel injection. Parameters are injected at multiple stages to improve conditioning. Physical coordinates are encoded to provide spatial information. This allows the model to dynamically adapt its computations to the specific PDE parameters governing each system, rather than learning a single fixed response. Experiments on the Advection-Diffusion-Reaction equation and Navier-Stokes flow around the cylinder wake demonstrate that our approach combines the efficiency of latent evolution with the fidelity of full-field models, outperforming DL-ROMs, latent transformers, and plain ViTs in multi-field prediction, reducing the relative rollout error by approximately $5$ times.

cs.LG↗

arXiv2025

Reduced Order Modeling of Partial Differential Equations on Parameter-Dependent Domains Using Deep Neural Networks

Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the equations over multiple parameters. Performing an exhaustive search over the parameter space requires solving the PDE multiple times, which is generally impractical. To address this challenge, reduced order models (ROMs) are built using a set of precomputed solutions (snapshots) corresponding to different parameter values. Recently, Deep Learning ROMs (DL-ROMs) have been introduced as a new method to obtain ROM, offering improved flexibility and performance. In many cases, the domain on which the PDE is defined also varies. Capturing this variation is important for building accurate ROMs but is often difficult, especially when the domain has a complex structure or changes topology. In this paper, we propose a Deep-ROM framework that can automatically extract useful domain parametrization and incorporate it into the model. Unlike traditional domain parameterization methods, our approach does not require user-defined control points and can effectively handle domains with varying numbers of components. It can also learn from domain data even when no mesh is available. Using deep autoencoders, our approach reduces the dimensionality of both the PDE solution and the domain representation, making it possible to approximate solutions efficiently across a wide range of domain shapes and parameter values. We demonstrate that our approach produces parametrizations that yield solution accuracy comparable to models using exact parameters. Importantly, our model remains stable under moderate geometric variations in the domain, such as boundary deformations and noise - scenarios where traditional ROMs often require remeshing or manual adjustment.

math.NA↗

arXiv2025

Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums

Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning -- specifically deep neural networks -- to learn optimal, potentially conductor-dependent weightings for Mestre--Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods.

math.NT↗

arXiv2022

Ranks of elliptic curves and deep neural networks

Determining the rank of an elliptic curve E/Q is a hard problem, and in some applications (e.g. when searching for curves of high rank) one has to rely on heuristics aimed at estimating the analytic rank (which is equal to the rank under the Birch and Swinnerton-Dyer conjecture). In this paper, we develop rank classification heuristics modeled by deep convolutional neural networks (CNN). Similarly to widely used Mestre-Nagao sums, it takes as an input the conductor of E and a sequence of normalized a_p-s (where a_p=p+1-#E(F_p) if p is a prime of good reduction) in some range (p<10^k for k=3,4,5), and tries to predict rank (or detect curves of ``high'' rank). The model has been trained and tested on two datasets: the LMFDB and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to 10^30, and rank up to 10. For comparison, eight simple neural network models of Mestre-Nagao sums have also been developed. Experiments showed that CNN performed better than Mestre-Nagao sums on the LMFDB dataset (interestingly neural network that took as an input all Mestre-Nagao sums performed much better than each sum individually), while they were roughly equal on custom made dataset.

math.NT↗

arXiv2022

Solving Bilevel Optimal Bidding Problems Using Deep Convolutional Neural Networks

Current state-of-the-art solution techniques for solving bilevel optimization problems either assume strong problem regularity criteria or are computationally intractable. In this paper we address power system problems of bilevel structure, commonly arising after the deregulation of the power industry. Such problems are predominantly solved by converting the lower-level problem into a set of equivalent constraints using the Karush-Kuhn-Tucker optimality conditions at an expense of binary variables. Furthermore, in case the lower-level problem is nonconvex, the strong duality does not hold rendering the single-level reduction techniques inapplicable. To overcome this, we propose an effective numerical scheme based on bypassing the lower level completely using an approximation function that replicates the relevant lower level effect on the upper level. The approximation function is constructed by training a deep convolutional neural network. The numerical procedure is run iteratively to enhance the accuracy. As a case study, the proposed method is applied to a price-maker energy storage optimal bidding problem that considers an AC power flow-based market clearing in the lower level. The results indicate that greater actual profits are achieved as compared to the less accurate DC market representation.

eess.SY↗

arXiv2022

Reconstruction of Incomplete Wildfire Data using Deep Generative Models

We present our submission to the Extreme Value Analysis 2021 Data Challenge in which teams were asked to accurately predict distributions of wildfire frequency and size within spatio-temporal regions of missing data. For the purpose of this competition we developed a variant of the powerful variational autoencoder models dubbed the Conditional Missing data Importance-Weighted Autoencoder (CMIWAE). Our deep latent variable generative model requires little to no feature engineering and does not necessarily rely on the specifics of scoring in the Data Challenge. It is fully trained on incomplete data, with the single objective to maximize log-likelihood of the observed wildfire information. We mitigate the effects of the relatively low number of training samples by stochastic sampling from a variational latent variable distribution, as well as by ensembling a set of CMIWAE models trained and validated on different splits of the provided data. The presented approach is not domain-specific and is amenable to application in other missing data recovery tasks with tabular or image-like information conditioned on auxiliary information.

stat.ML↗

arXiv2021

The box dimension of degenerate spiral trajectories of a class of ordinary differential equations

In this paper we initiate the study of the box dimension of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type $(n,n)$ by exploiting the well-known fractal results for $α$-power spirals. In the general $(m,n)$ case, we formulate a conjecture about the box dimension of a degenerate focus. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their "entry-exit" function. There exists a bijective correspondence between the box dimension of the slow-fast spiral and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.

math.DS↗

arXiv2020

BlackBox: Generalizable Reconstruction of Extremal Values from Incomplete Spatio-Temporal Data

We describe our submission to the Extreme Value Analysis 2019 Data Challenge in which teams were asked to predict extremes of sea surface temperature anomaly within spatio-temporal regions of missing data. We present a computational framework which reconstructs missing data using convolutional deep neural networks. Conditioned on incomplete data, we employ autoencoder-like models as multivariate conditional distributions from which possible reconstructions of the complete dataset are sampled using imputed noise. In order to mitigate bias introduced by any one particular model, a prediction ensemble is constructed to create the final distribution of extremal values. Our method does not rely on expert knowledge in order to accurately reproduce dynamic features of a complex oceanographic system with minimal assumptions. The obtained results promise reusability and generalization to other domains.

cs.LG↗

arXiv2017

A note on the box dimension of degenerate foci

We study polynomial planar systems with singularity of focus type without characteristic directions. Simple and natural transformation of weak focus has been used to obtain such degenerate focus. We compute the box dimension of a spiral trajectory, and show connection to cyclicity of the system under a perturbation.

math.DS↗

arXiv2016

Oscillatory Integrals and Fractal Dimension

We study geometrical representation of oscillatory integrals with an analytic phase function and a smooth amplitude with compact support. Geometrical properties of the curves defined by the oscillatory integral depend on the type of a critical point of the phase. We give explicit formulas for the box dimension and the Minkowski content of these curves. Methods include Newton diagrams and the resolution of singularities.

math.CA↗

arXiv2014

Wavy spirals and their fractal connection with chirps

We study the fractal oscillatority of a class of real $C^1$ functions $x=x(t)$ near $t=\infty$. It is measured by oscillatory and phase dimensions, defined as box dimensions of the graph of $X(τ)=x(\frac{1}τ)$ near $τ=0$ and the trajectory $(x,\dot{x})$ in $\mathbb{R}^2$, respectively, assuming that $(x,\dot{x})$ is a spiral converging to the origin. The relationship between these two dimensions has been established for a class of oscillatory functions using formulas for box dimensions of graphs of chirps and nonrectifiable wavy spirals, introduced in this paper. Wavy spirals are a specific type of spirals, given in polar coordinates by $r=f(φ)$, converging to the origin in non-monotone way as a function of $φ$. They emerged in our study of phase portraits associated to solutions of Bessel equations. Also, the rectifiable chirps and spirals have been studied.

math.CA↗

arXiv2014

Geometrical properties of a class of systems with spiral trajectories in R^3

Here we study a class of second-order nonautonomous differential equations, and the corresponding planar and spatial systems, from the point of view of fractal geometry. The fractal oscillatority of solutions at infinity is measured by oscillatory and phase dimensions. The oscillatory dimension is defined as the box dimension of the reflected solution near the origin, while the phase dimension is defined as the box dimension of a trajectory of the corresponding planar system in the phase plane. Using the phase dimension of the second-order equation we compute the box dimension of a spiral trajectory of the spatial system, lying in Lipschitzian or H\" olderian surfaces. This phase dimension of the second-order equation is connected to the asymptotics of the associated Poincaré map. Also, the box dimension of a trajectory of the reduced normal form with one eigenvalue equals to zero, and a pair of pure imaginary eigenvalues has been computed when limit cycles bifurcate from the origin.

math.CA↗

arXiv2013

Fractal properties of Bessel functions

A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory $(x,\dot{x})$ in $\mathbb{R}^2$ of a solution $x=x(t)$, assuming that $(x,\dot{x})$ is a spiral converging to the origin. In this work, we study the phase dimension of the class of second-order nonautonomous differential equations with oscillatory solutions including the Bessel equation. We prove that the phase dimension of Bessel functions is equal to 4/3, and that the corresponding trajectory is a wavy spiral, exhibiting an interesting behavior. The phase dimension of a generalization of the Bessel equation has been also computed.

math.CA↗