Program

We use sigmoidal random feature maps to discern the Hamiltonian and symplectic structure from vector field data. This allows us to infer the ‘nearest’ Hamiltonian system to given data and also immediately yields a symplectic splitting method to make predictions of the evolution of that system. The approach works for any nondegenerate symplectic structure. I will also discuss extensions to systems with Lie-Poisson and affine Poisson structure.

In his 1982 thesis, Jens Hoppe discovered what became known as the \(SU(k) -> SDiff(S^2)\) limit: commutation relations between square matrices of size k converge, in the limit of large k, to the Lie algebra of the group of volume-preserving diffeomorphisms of the 2-sphere. In subsequent work by Bordemann, Schlichenmaier, Meinrenken and others in the late 1980s and early 1990s, this curious observation was explained in terms of geometric quantization, and it has found many uses – from fuzzy sphere models in noncommutative geometry to Zeitlin’s matrix approach in hydrodynamics. In a parallel but unrelated development, the representation theory for centrally extended loop groups \(LSU(k)\) (Kac-Moody algebras) revolutionised the way conformal field theories in 1+1 dimension were handled in string theory and condensed matter theory. In this talk, based on joint work with Zhenghan Wang (Microsoft station Q/UCSB), we investigate what happens if you combine these developments. We show that the Kac-Moody cocycles on \(LSU(k)\) have `fuzzy sphere limits’ in \(LSDiff(S^2)\), and discuss some conjectures on what this might entail for conformal field theories in 2+1 dimensions.

Among the advances of Isogeometric Analysis is the ability to simulate higher-order problems by using spline bases of higher-order continuity. This particularly benefits phase-field problems such as the simulation of the Cahn-Hilliard equation, e.g., for the simulation of phase-field fracture. Another benefit of Isogeometric Analysis, in particular of nested spline spaces, is the ability to represent spline derivatives exactly through, for example, De Rham complexes, enabling also exact representation of quantities like the Jacobian determinant.

Even though IGA has excellent convergence properties, hence its great efficiency, a large number of repeated Full-Order Model computations over a parametric space can still be a computationally infeasible task. Therefore, Reduced-Order Modeling (ROM) techniques are required in settings such as Uncertainty Quantification or Bayesian Inversion, where the number of model evaluations can be several orders of magnitude larger. Recent advances in ROM include Operator Learning, where one attempts to develop a ROM based on learning an operator structure, potentially on a (reduced) discrete space. Examples include Operator Inference (OpInf) [1], IGANets [2] and Neural Green’s Operators (NGOs) [3], among other methods. The challenge, however, is to model non-linear operators accurately.

Through lifting, where snapshots are augmented with auxiliary fields to expose a polynomial or linear structure to the operator, Operator Inference methods can be trained more accurately, and, for example, conservation laws can be embedded in the operator. Since Isogeometric Analysis can provide exact representation of derivative fields, or more generally polynomials of those, it directly benefits Operator Learning methods which support lifting. For example, operator inference for the Cahn-Hilliard equation can potentially be improved by lifting the solution coefficients, and the coefficients corresponding to the gradient and Laplacian fields, which can be generated with arbitrary smooth spline bases. In case of standard Finite Element Analysis, construction of discrete derivatives is, instead, a non-trivial task. In this talk, we provide results into an investigation of Operator Learning via Operator Inference. Aiming to construct Reduced-Order Models for problems such as the Cahn-Hilliard equation, we investigate how operator inference can benefit from the lifting of exactly represented quantities represented on spline bases.

1. Peherstorfer, Benjamin, and Karen Willcox. “Data-Driven Operator Inference for Nonintrusive Projection-Based Model Reduction.” Computer Methods in Applied Mechanics and Engineering 306 (July 2016): 196–215. https://doi.org/10.1016/j.cma.2016.03.025.
2. Möller, Matthias, Günther Obermair, Isabella Singer, Christian Gollmann, Alessandro Reali, and Stefanie Elgeti. “IGANets: Isogeometric Analysis Networks and Their Applications to Linear Structural Analysis Problems.” Engineering with Computers 42, no. 3 (2026): 102. https://doi.org/10.1007/s00366-026-02312-6.
3. Melchers, H. A., J. H. M. Prins, and M. R. A. Abdelmalik. “Neural Green’s Operators for Parametric Partial Differential Equations.” Computer Methods in Applied Mechanics and Engineering 455 (June 2026): 118893. https://doi.org/10.1016/j.cma.2026.118893.

Since its introduction in 1963, large-eddy simulation (LES) has advanced significantly and has developed into a valuable tool to study turbulent flow. LES resolves only motions larger than a prescribed length scale and models the combined (dissipative) effect of the unresolved motions. The model for unresolved small-scale turbulence and the numerical discretization of the governing equations are often treated as modular components, each replaceable by alternatives with equivalent functionality. However, this modularity overlooks the complex nonlinear interactions between modeling and discretization errors, which remain poorly understood yet critically influence overall simulation accuracy. In practice, the boundaries between filtering, modeling and discretization become blurred, as numerical schemes influence both the effective filter and the modeled dissipation. This work therefore treats the core elements of LES as a unified whole; it seeks to advance LES through the integration of filtering, modeling and discretization. The integrated approach is developed and demonstrated for relatively simple turbulent flows, serving as an initial step toward future extensions to complex turbulent flows.

Structure-preserving discretizations have proven to enhance the both the stability and physical reliability of the numerical simulation of continuum mechanics, and fluid mechanics in particular (e.g.: the DNS of turbulent flows). The construction of structure-preserving discretizations proceeds flawlessly into an existing, underlying mesh representing the domain of interest by mimicking Hodge complexes (exterior derivatives and Hodge stars) and Cartan calculus (Lie derivatives) into their discrete counterparts. However, in the context of multiphase flows, the domain is partitioned into different phase regions (e.g.: liquid and gas) and its representation is not trivial anymore. While body-fitted approaches exists, in this talk we explore structure-preserving discretizations when the phases are embedded into a fixed domain, i.e.: the background mesh is kept fixed and the interface is represented implicitly, with a level-set like function. In particular, we look at the discretization of surface tension, a well-know troublemaker term, in the context of the Navier-Stokes equations for bubble and droplet flows and discuss the conservation of both linear momentum and kinetic energy of such a term.

Structure-preserving discretisations aim to retain, at the discrete level, fundamental invariants and geometric structures of partial differential equations (PDEs). Typical examples include conservation of energy, momentum, and helicity for the Navier–Stokes equations, and potential enstrophy for the shallow-water equations. Beyond their conceptual relevance, such discretisations often lead to concrete numerical advantages, including enhanced stability, the elimination of spurious modes, and improved long-time accuracy.

The Finite Element Exterior Calculus (FEEC) framework provides a rigorous mathematical foundation for the construction of structure-preserving finite element discretisations. FEEC is based on the identification and discretisation of Hilbert complexes underlying the continuous problem, such as the de Rham complex — with applications in electromagnetics or diffusion problems — and the elasticity complex. Its formulation relies heavily on differential-geometric concepts: physical fields are represented as differential forms, differential operators are unified through the exterior derivative, and constitutive and metric relations are encoded via Hodge-⋆ operators.

Most existing finite element libraries are built around classical vector-calculus formulations and incorporate FEEC concepts indirectly. While effective, this approach often requires ad hoc constructions to recover the underlying geometric structure, and can complicate the generalisation to higher dimensions, non-standard function spaces, or complex geometries.

Mantis.jl is a Julia finite element library designed natively around the FEEC paradigm. It provides a flexible environment for formulating and discretising PDEs directly in the language of exterior calculus, supporting arbitrary spatial dimensions and a wide range of discrete function spaces.

Lie groups and their actions are ubiquitous in the description of physical systems. In recent work, we explored implications in the setting of model order reduction (MOR). The resulting approach of MOR via Lie groups (MORLie) abstracts full order models (FOMs) as high-dimensional dynamical systems on manifolds, and reduced order models (ROMs) as low-dimensional dynamical systems on Lie groups. The approach extended existing Lie group frameworks for MOR to non-equivariant dynamics, which are frequent in practical applications, lent itself to non-intrusive MOR methods that strongly outscale linear subspace methods, and has an error bound lower than the Kolmogorov N-width. In this talk, we present a first extension of MORLie to port-Hamiltonian systems on manifolds, which capture the energy-routing and dissipative structure of a vast number of controlled physical systems.

In this talk I present a boundary-value-method (BVM) based on a higher-order implicit midpoint method. This class of methods usually applies an additional final condition to close the discretized system. BVM’s offer superior stability properties compared to classical initial-value multistep methods, especially for stiff and semi-stable differential equation systems. Since information is coupled over the whole integration interval, these methods can achieve large regions of absolute stability and reduce parasitic error growth. In particular, high-order boundary-value methods may retain excellent stability characteristics where standard BDF or Adams-type schemes lose robustness. Unstable or ill-posed DEs, such as the backward heat equation or the backward SIR-model, may be resolved numerically as well, due to this property. Here, we focus on periodic solutions in Hamiltonian systems, where the final condition is replaced by a periodic condition, as suggested by Axelsson and Verwer in 1986. Several examples will be given to illustrate the potential of BVM’s.

When simulating flow instabilities, as in turbulent flow or at the tip of breaking waves, it is essential that these are not suppressed by non-physical, numerical dissipation. This implies that the discretization of convection should be free of artificial viscosity. Also, the pressure should only do work related to compression. Following the principle of non-interference, only molecular viscosity is allowed to remove energy from the flow. In the presentation we will focus on the convective term, and discuss necessary and sufficient conditions for satisfying the above requirements for a large family of discretization methods.

This paper considers structure-preserving model order reduction (MOR) techniques for port- Hamiltonian (pH) systems, which are typically derived from energy-based modelling. To keep favorable properties of pH systems such as stability and passivity in a reduced order model (ROM), we use structure-preserving methods in the reduction process. There exists an extensive literature on structure-preserving MOR methods of pH systems, however, to the best of our knowledge, there does not exist an intrusive structure-preserving MOR method for nonlinear pH systems on the base of general nonlinear approximation maps. To close this gap, we propose a MOR method for pH systems based on the idea of the generalized manifold Galerkin (GMG) reduction. The resulting MOR method can be applied to both linear and nonlinear pH systems resulting in ROMs, which are again of pH form. For the numerical examples, we employ a linear and a nonlinear mass-spring-damper system and the results show that the proposed MOR methods have lower relative reduction error compared to existing methods.

Geometric considerations have proven to be very beneficial in structure-preserving high-order methods in space. Similar considerations can be applied in the time domain as well. The construction of such methods will be presented and its properties discussed.

The construction of structure-preserving discretizations requires careful attention to the choice of PDE formulation: it is well known that discretizing different, but analytically equivalent, formulations of the same PDE can yield discrete systems with vastly different conservation properties. In this presentation, we discuss how the Hamiltonian formulation of the shallow water equations provides a principled route to the correct formulation. Specifically, the Hamiltonian structure singles out the rotational form of the convective term and introduces auxiliary variables, and this combination, when discretized within the finite element exterior calculus (FEEC) framework, yields a scheme that conserves both energy and potential vorticity exactly at the discrete level.