arXiv subjects

Yurij Salmaniw

Publications and source records attributed to Yurij Salmaniw.

9 recordsLinked to original sources

arXiv2026

Learning functional components of PDEs from data using neural networks

Partial differential equation (PDE) models frequently contain unknown functional terms that cannot be measured directly, limiting their predictive utility. While data-driven methods for estimating scalar PDE parameters are well established, the recovery of unknown functions remains comparatively underexplored. Here, we show that standard parameter estimation workflows can be extended to infer functional components of PDEs directly from data. Our approach embeds neural networks within the PDE framework, allowing unknown functions to be learned during training with high accuracy. Using nonlocal aggregation-diffusion equations as a case study, we infer interaction kernels and external potentials from steady-state observations. We systematically examine how reconstruction accuracy depends on factors such as the number and diversity of available solutions, sampling density, and measurement noise. The resulting framework retains the advantages of conventional PDE calibration approaches while extending them to functional inference: once trained, the PDE model can be used in the standard way to analyse system behaviour and generate predictions.

cs.LG↗

arXiv2026

Numerical stationary states for nonlocal Fokker-Planck equations via fixed points of consistency maps

We propose a fixed-point-based numerical framework for computing stationary states of nonlocal Fokker-Planck-type equations. Instead of discretising the differential operators directly, we reformulate the stationary problem as a nonlinear fixed-point map built from the original PDE and its nonlocal interaction terms, and solve the resulting finite-dimensional problem with a matrix-free Newton-Krylov method. We compare implementations using the analytic Frechet derivative of this map with a simple central-difference approximation. Because the method does not rely on time evolution, it is agnostic to dynamical stability and can detect both stable and unstable stationary states. Its accuracy is determined mainly by the numerical treatment of convolutions and quadrature, rather than by differentiation stencils. We apply the approach to three model problems with linear diffusion, use existing analytical results to verify the outputs, and reproduce known bifurcation diagrams, as well as new bifurcation behaviour not previously observed in this kind of problem.

math.NA↗

arXiv2025

Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus

We investigate stationary states, including their existence and stability, in a class of nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions. For the scalar case, we extend previous results by showing that key model features, such as existence, regularity, bifurcation structure, and stability exchange, continue to hold under a mere bounded variation hypothesis. For the corresponding two-species system, we carry out a fully rigorous bifurcation analysis using the bifurcation theory of Crandall & Rabinowitz. This framework allows us to classify all solution branches from homogeneous states, with particular attention given to those arising from the self-interaction strength and the cross-interaction strength, as well as the stability of the branch at a point of critical stability. The analysis relies on an equivalent classification of solutions through fixed points of a nonlinear map, followed by a careful derivation of Fr\'echet derivatives up to third order. An interesting application to cell-cell adhesion arises from our analysis, yielding stable segregation patterns that appear at the onset of cell sorting in a modelling regime where all interactions are purely attractive.

math.AP↗

arXiv2025

Dynamics of a coupled nonlocal PDE-ODE system with spatial memory: well-posedness, stability, and bifurcation analysis

Nonlocal aggregation-diffusion models, when coupled with a spatial map, can capture cognitive and memory-based influences on animal movement and population-level patterns. In this work, we study a one-dimensional reaction-diffusion-aggregation system in which a population's spatiotemporal dynamics are tightly linked to a separate, dynamically updating map. Depending on the local population density, the map amplifies and suppresses certain landscape regions and contributes to directed movement through a nonlocal spatial kernel. After establishing the well-posedness of the coupled PDE-ODE system, we perform a linear stability analysis to identify critical aggregation strengths. We then perform a rigorous bifurcation analysis to determine the precise solution behavior at a steady state near these critical thresholds, deciding whether the bifurcation is sub- or supercritical and the stability of the emergent branch. Based on our analytical findings, we highlight several interesting biological consequences. First, we observe that whether the spatial map functions as attractive or repulsive depends precisely on the map's relative excitation rate versus adaptory rate: when the excitatory effect is larger (smaller) than the adaptatory effect, the map is attractive (repulsive). Second, in the absence of growth dynamics, populations can only form a single aggregate. Therefore, the presence of intraspecific competition is necessary to drive multi-peaked aggregations, reflecting higher-frequency spatial patterns. Finally, we show how subcritical bifurcations can trigger abrupt shifts in average population abundance, suggesting a tipping-point phenomenon in which moderate changes in movement parameters can cause a sudden population decline.

math.AP↗

arXiv2024

Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators

Parameter identifiability is often requisite to the effective application of mathematical models in the interpretation of biological data, however theory applicable to the study of partial differential equations remains limited. We present a new approach to structural identifiability analysis of fully observed parabolic equations that are linear in their parameters. Our approach frames identifiability as an existence and uniqueness problem in a closely related elliptic equation and draws, for homogeneous equations, on the well-known Fredholm alternative to establish unconditional identifiability, and cases where specific choices of initial and boundary conditions lead to non-identifiability. While in some sense pathological, we demonstrate that this loss of structural identifiability has ramifications for practical identifiability; important particularly for spatial problems, where the initial condition is often limited by experimental constraints. For cases with nonlinear reaction terms, uniqueness of solutions to the auxiliary elliptic equation corresponds to identifiability, often leading to unconditional global identifiability under mild assumptions. We present analysis for a suite of simple scalar models with various boundary conditions that include linear (exponential) and nonlinear (logistic) source terms, and a special case of a two-species cell motility model. We conclude by discussing how this new perspective enables well-developed analysis tools to advance the developing theory underlying structural identifiability of partial differential equations.

math.AP↗

arXiv2024

Well-posedness of aggregation-diffusion systems with irregular kernels

We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential $K$. We are interested in establishing their well-posedness theory when the nonlocal interaction potential $K$ is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that $\nabla K\ast K$ is in $L^2$, we can prove that the solution is smooth and unique. When $K$ is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the $n$-species system.

math.AP↗

arXiv2024

From habitat decline to collapse: a spatially explicit approach connecting habitat degradation to destruction

Habitat loss, driven primarily by anthropogenic activity, significantly threatens ecosystem sustainability. While it is well understood that habitat loss is the leading contributor to declines in biodiversity worldwide, the connection between habitat degradation, destruction, and different locomotion strategies remains unclear. We use a reaction-diffusion framework to analyze the effects of habitat loss on population persistence and abundance. We establish necessary and sufficient conditions for the existence of an extinction threshold, beyond which further degradation of the environment predicts deterministic extirpation. Our results offer a robust analytical connection between habitat degradation and destruction, providing a mechanistic understanding of species persistence under varying environmental conditions and differing locomotion strategies.

math.AP↗

arXiv2023

Biological Aggregations from Spatial Memory and Nonlocal Advection

We investigate a nonlocal single-species reaction-diffusion-advection model that integrates the spatial memory of previously visited locations and nonlocal detection in space, resulting in a coupled PDE-ODE system reflective of several existing models found in spatial ecology. We prove the existence and uniqueness of a Hölder continuous weak solution in one spatial dimension under some general conditions, allowing for discontinuous kernels such as the top-hat detection kernel. A robust spectral and bifurcation analysis is also performed, providing the rigorous analytical study not yet found in the existing literature. In particular, the essential spectrum is shown to be entirely negative, and we classify the nature of the bifurcation near the critical values obtained via a linear stability analysis. A pseudo-spectral method is used to solve and plot the steady states near and far away from these critical values, complementing the analytical insights.

math.AP↗

arXiv2023

Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping

The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose numerous open problems relevant to this relatively new area, ranging from precise technical mathematical challenges to broader conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.

math.AP↗