The perspective of pattern formation has been successful in drawing from and helping advance multiple areas of mathematics, including dynamical systems, partial differential equations and numerical computing. Formal asymptotic and rigorous approaches such as spatial dynamics have been highly successful over the past years to study/prove the existence and stability of patterns in one spatial dimension. They have also been extended to higher dimensions under certain geometries: such as cylinderical, channel-like domains, etc. They are also useful in understanding invasion fronts, localised patterns, spiral waves and defects in 1D. However, the extension of the wealth of the above mentioned approaches to the analysis of patterns in 2D/3D is not straightforward. 

 

A non-exhaustive list of examples of situations that have proved to be resistant to analysis, and yet very relevant in diverse applications are: patterns formed with more than one preferred lengthscale, aperiodic patterns, multi-dimensional defects, spatial localisation without radial symmetry, patterns in heterogeneous domains, patterns in the presence of a dynamic bifurcation parameter, patterns in lattice systems and non-local systems. However in all of these examples, we are able to obtain numerical approximations to equilibria of the associated governing PDE, either through an initial-boundary value problem approach (time-stepping) or via a root-finding approach (numerical continuation). 

 

Since it is a non-objective function if numerical computability equals proof of existence, I want to explore novel and dimensionally agnostic, intra-disciplinary bridges to pattern formation, that will help us to obtain (using computational algebraic geometry), analyse (using computer assisted proofs as a certification problem) and characterise (using topological data analysis) truly multi-dimensional patterns. 

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Title: (Georgina) Modelling intermediate-current transitions in asymmetric-valence binary electrolytes

Abstract: The valences of ions in a binary electrolyte impact the performance of electrochemical devices, but most electrochemical modelling focuses on symmetric 𝑧 :𝑧 binary electrolytes. We study the impact of asymmetric ion valences on the spatial distribution of the positive and negative ion concentrations and electric potential inside a simple electrochemical device. We consider a one-dimensional steady-state Poisson–Nernst–Planck model with imposed constant ionic fluxes. Numerical simulations reveal a smooth valence-dependent transition point at an intermediate current where the classical boundary layers vanish. We fully characterise this transition using asymptotic analysis. In addition, we produce implicit analytic expressions for general asymmetric binary electrolytes alongside explicit solutions for 2⁢𝑧 :𝑧, 𝑧 :2⁢𝑧, and symmetric 𝑧 :𝑧 electrolytes. Our results collapse onto a suitably scaled phase diagram to predict the observed transition in terms of ion valences and fluxes.

 

 

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Title: (Callum) Extended Pseudo-spectral Physics-informed Neural Networks for Phase-field Models

Abstract: Phase-field models provide a fundamental continuum framework for describing phase separation and pattern formation in many physical and biological systems. Their predictive capability depends critically on constitutive quantities such as the bulk free-energy density and interfacial thickness parameter, which are often unknown and must be inferred from limited observations. In this work, we introduce an extended pseudo-spectral physics-informed neural network (ESPINN) framework for the inverse identification of phase-field models from transient snapshot data. The proposed method simultaneously reconstructs the bulk chemical potential and unknown gradient coefficients directly from dynamically evolving structures.

Numerical experiments show that ESPINN accurately recovers both the functional form of the free energy and the interfacial thickness parameter. Remarkably, substantial constitutive information can be extracted even from a single snapshot pair, while additional snapshots improve robustness and reduce variance across training runs. The framework remains stable in the presence of noise, with reconstruction accuracy improving as more observations are incorporated. These results highlight ESPINN as a data-efficient and physically consistent approach for learning constitutive structure in continuum models of phase separation.

 

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