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Organiser: Shreyas Mandre (from term 3)

The Applied Maths Seminars are held on Fridays 12:00-13.00. This year the seminar will be hybrid: you can choose to attend in person in room B3.02 or on MS Teams. The team for the seminar is the same as last year, but if you are not a member, you can send a membership request via MS Teams or email the organisers.

Please contact Shreyas Mandre if you have any speaker suggestions for term 3.

Seminar Etiquette: Here is a set of basic rules for the seminar.

  • Please keep your microphone muted throughout the talk. If you want to ask a question, please raise your hand and the seminar organiser will (a) ask you to unmute if you are attending remotely or (b) get the speaker's attention and invite you to ask your question if you are in the room.
  • If you are in the room with us, the room microphones capture anything you say very easily, and this is worth keeping in mind ☺️
  • You can choose to keep your camera on or not. Colleagues in the room will be able to see the online audience.
  • Please let me know if you would like to attend any specific talk in person and/or have lunch with any of the speakers who are coming to visit us so that I can make sure you have a place in the room.

Term 3

Term 2

Please note that we are still waiting for confirmation of restrictions on the new year to confirm whether our speakers will be able to come in person in term 2.

Term 1


Term 3

Week 1. Laura Cope (Leeds) - Unearthing the Dynamics of Planetary Jet Streams

Jet streams, or zonal jets, are strong and persistent east-west flows that arise spontaneously in planetary atmospheres and oceans. They are ubiquitous, with key examples including mid-latitude jets in the troposphere, multiple jets in the Antarctic Circumpolar Current and flows on gaseous giant planets such as Jupiter and Saturn. Turbulent flows on a beta-plane lead to the spontaneous formation and equilibration of persistent zonal jets. However, the equilibrated jets are not steady and the nature of the time variability in the equilibrated phase is of interest both because of its relevance to the behaviour of naturally occurring jet systems and for the insights it provides into the dynamical mechanisms operating in these systems. I will discuss aspects of zonal jet variability using insights from a framework of barotropic beta-plane models in which eddy-eddy interactions are systematically neglected.

Week 2. Greg Chini (U New Hampshire) - A Tale of Two Quasi-Linear Fluid Dynamical Systems: Modulated Waves and Shear-Driven Instabilities

The quasi-linear (QL) approximation has facilitated the prediction and understanding of a broad variety of fluid dynamical phenomena, ranging from the quasi-biennial oscillation of the zonal winds in the equatorial stratosphere to the emergence of exact coherent states (ECS) in wall-bounded turbulent shear flows. The QL reduction involves a decomposition into mean and fluctuation components and retention of fluctuation/fluctuation nonlinearities only where they feed back on the mean dynamics. Although sometimes invoked as an ad hoc simplification, the QL approximation can be justified asymptotically for certain flows exhibiting temporal scale separation, as will be demonstrated here through two complementary examples. In the first example, a new type of acoustically-driven mean flow is identified. It is shown that when a high-frequency acoustic wave of small amplitude \epsilon interacts with a stratified fluid, an unusually strong form of acoustic streaming can occur, with the time-mean flow arising at O(\epsilon) rather than the more commonly realized O(\epsilon^2) value. The resulting two-way coupling between the wave and streaming flow is self-consistently captured in a QL dynamical system. In the second illustration, a QL model of strongly stratified turbulent shear flows is derived. Spectrally non-local energy transfers, associated with small-scale non-hydrostatic instabilities induced by the relative horizontal motion of large-scale hydrostatic eddies, are economically represented. The model is used to compute ECS in a body-forced, strongly stratified flow and to evaluate the mixing efficiency achieved by these nonlinear states. For both the wave- and shear-driven systems, new asymptotic analyses are developed that enable integration of the dynamics strictly on the slow time scale associated with the mean flow, yielding significant computational efficiencies while simultaneously promoting physical insight.

Week 3. Srikanth Toppaladoddi (Leeds) - Nonlinear interactions between fluid flows and evolving boundaries

Interactions between fluid flows and evolving boundaries are at the heart of some of the most challenging problems in applied mathematics, geophysics, climate physics, and engineering. In this talk, I will explore the interplay between shear- and buoyancy-driven flows and phase-changing boundaries using numerical simulations. Specifically, I will consider the Rayleigh-Bénard-Poiseuille flow of a pure liquid phase over its solid phase to understand the impact of those interactions on the evolution of the solid phase and the heat transport in the liquid phase.

Week 4. Marcelo Dias (Edinburgh) - On kirigami mechanics

Kirigami, the Japanese art of paper cutting, offers unexplored ways to tailor the morphology of thin elastic sheets. By exploiting the fundamental principles of this art and carefully tuning the geometry and the topology of the cuts, it unlocks a great potential to control the structural and mechanical properties of thin sheets across multiple scales. In this talk, we will explore the emergent local non-linear effects in Kirigami by focusing our attention on the study of the deformation of a thin sheet with a single cut, the most fundamental deformation motif. We will consider its deformation following the opening of the slit by a given excess angle. In the isometric limit the elastic energy has no stretching contribution and the shape of the disk is governed by the bending energy as it approaches that of an e-cone: a conical solution where all the generators remain straight and intersect at a singularity on its apex. We will show how we solved the geometrically nonlinear boundary value problem for a Saint Venant-Kirchhoff constitutive plate model to find the geometry of the e-cone as well as the surface stresses and couple-stresses (moments). We will further investigate a model where stretching energy plays a role in determining the state of the fundamental motif.

Week 5. Anagha Madhusudanan (Cambridge) - Coherent structures from the linearized Navier–Stokes equations for wall-bounded turbulent flows

We study turbulent wall-bounded flows using a linearized Navier-Stokes equations based model. To obtain this model, the non-linearity in the Navier-Stokes equations are isolated as a disturbance term. The disturbance is then considered to be a forcing to the linearized equations (e.g. McKeon & Sharma, 2010, Hwang & Cossu, 2010 etc.). This rearrangement of the equations gives us a linear model that can then be studied using standard tools. From the literature we know that these linearized equations model structures that are reminiscent of the large-scale coherent structures found in turbulent wall-bounded flows. This observation is significant because such large-scale structures play crucial roles in these flows. In this talk we will concentrate on these large-scale structures that are modelled by the linearized Navier–Stokes equations in a turbulent channel flow. We consider three separate aspects of these structures: i) their wall-normal coherence, ii) the estimation of their instantaneous and statistical features and iii) the effect of stratification on them. In all three cases the trends from the model are compared to direct numerical simulation datasets.

Week 6: No seminar

Week 7: Meritxell Saez (Crick Institute): Modelling of cell decision-making using catastrophe theory and dynamical systems

The generation of cellular diversity during development, from zygote to embryo to a fully formed individual, involves differentiating cells transitioning between discrete cell states. The developmental path of a cell can be modelled as a trajectory that runs through several attractors in a dynamical system that changes due to external perturbations. Decisions take place when the trajectory is close to a saddle point. The dynamical systems defined by the potentials around the elementary catastrophes correspond to different decision-making scenarios and allow us to organise and classify the possible biological mechanisms. The models that arise provide the foundations for quantitative geometric models of cellular differentiation and can be fit to experimental data and used to make quantitative predictions about cellular differentiation.

Week 8: Alethea Barbaro (Delft): TBC
Week 9: Anna Lisa Varri (Edinburgh): TBC
Week 10: Jake Langham (Bristol): TBC

Term 2

Week 1. Draga Pihler-Puzovic (Manchester) - Problems in viscous fingering

Growth of complex fingers at the interface of air and a viscous fluid in the narrow gap between two parallel plates is an archetypical problem of pattern formation. This problem is quasi-two-dimensional and its key information is encapsulated in the interface, so it is relatively simple to study both experimentally and theoretically compared to many other nonlinear phenomena. In this talk we consider two variants of this system that allows us to ask different questions, one of the fundamental nature and one of practical importance.

Firstly, we consider the fingering between plates which are subject to a time-dependent (power-law) plate separation. We demonstrate that in this system the interface can evolve in a self-similar fashion such that the interface shape at a given time is simply a rescaled version of the shape at an earlier time. These novel, self-similar solutions are linearly stable but they only develop if the initially circular interface is subjected to unimodal perturbations. Conversely, the application of nonunimodal perturbations (e.g., via the superposition of multiple linearly unstable modes) leads to the development of complex, constantly evolving finger patterns similar to those that are typically observed between rigid stationery plates. We discuss these finding in the context of understanding disordered front propagation more generally.

Secondly, we demonstrate that the lubrication flow in the narrow gap between boundaries, in which one of the rigid plates has been replaced by a confined elastic solid, can `choke' at high flow rates, due to the deforming solid making contact with the plate and sealing the gap. This phenomenon is important in the context of designing soft microfluidics and predicting behaviour of deformable porous media. We show that, depending on the finger morphology, the fingering instability can either promote or suppress the choking in experiments compared with analogous axisymmetric simulations at the same control parameters. We discuss physical mechanisms that lead to this behaviour.

Week 2. Balasz Kovacs (Regensburg) - A convergent algorithm for the interaction of mean curvature flow and diffusion

In this talk we will present an evolving surface finite element algorithm for the interaction of forced mean curvature flow and a diffusion process on the surface.
The evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by the above coupled geometric PDE system. The coupled system is inspired by the gradient flow of a coupled energy, we will use this model for introductory purposes.
We will present two algorithms, based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface.
For one of the numerical methods we will give some insights into the stability estimates which are used to prove optimal-order $H^1$-norm error estimates for finite elements of polynomial degree at least two.
We will present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
The talk is based on joint work with C.M. Elliott (Warwick) and H. Garcke (Regensburg).

Week 3. Mark Opmeer (Bath) - Model Reduction and the Numerical Solution of Lyapunov Equations

We will consider Model Reduction in the context of partial differential equations where the aim is to approximate the PDE by a (low-dimensional) system of ODEs with little loss of accuracy.

Based on singular value analysis of an associated Hankel operator, we will show when this is possible and when it is not (essentially: it is possible for parabolic PDEs and isn't possible for hyperbolic PDEs).

The actual computation of the reduced order model requires the numerical solution of an (operator) Lyapunov equation. We will analyse a numerical algorithm which efficiently does this.

Week 4. Nastassia Pouradier Duteil (Inria Paris) - Continuum limits of collective dynamics with time-varying weights

In this talk, we will derive the mean-field limit of a collective dynamics model with time-varying weights. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution and introduce a new empirical measure (in the position space) taking into account the weights. Continuity with respect to the initial data allows us to prove the convergence of the microscopic system to the macroscopic equation. This mean-field limit can be derived if the particles' dynamics preserve indistinguishability. If they do not, another point of view consists of deriving the so-called graph limit. We will introduce the graph limit and show the subordination of the mean-field limit to the graph limit equation.

Week 5. Marta Catalano (Warwick Statisics) - A Wasserstein index of dependence for Bayesian nonparametric modeling

Optimal transport (OT) methods and Wasserstein distances are flourishing in many scientific fields as an effective means for comparing and connecting different random structures. In this talk we describe the first use of an OT distance between Lévy measures with infinite mass to solve a statistical problem. Complex phenomena often yield data from different but related sources, which are ideally suited to Bayesian modeling because of its inherent borrowing of information. In a nonparametric setting, this is regulated by the dependence between random measures: we derive a general Wasserstein index for a principled quantification of the dependence gaining insight into the models’ deep structure. It also allows for an informed prior elicitation and provides a fair ground for model comparison. Our analysis unravels many key properties of the OT distance between Lévy measures, whose interest goes beyond Bayesian statistics, spanning to the theory of partial differential equations and of Lévy processes.

Week 6. Adam Townsend (Durham) - Microorganisms swimming through structured networks: from the point of view of the microorganism

Microorganisms swimming through viscoelastic fluids are a common feature in naturally-occurring fluids. Approaches to modelling this behaviour normally come in two flavours: a macroscale, course-grained approach where the fluid is modelled as a continuum; and a microscale, detailed approach where the anisotropic structure within the background fluid is specifically modelled. The former approach has the benefit of speed, but in this talk I will share my experience of attempting the latter using computer simulations, and show how these simulations suggest the physical structure of the viscoelastic fluid directly affects the microorganism swimming through it.

Week 7. Ben Goddard (Edinburgh) - Bounded Confidence Models of Opinion Dynamics

Mathematical modelling in the social sciences has a long and varied history, and has recently attracted increased attention. In addition to earlier, and by now well established, areas such as traffic flow, crowd dynamics, and urban modelling, new applications of mathematics to the social sciences have emerged. One such application is the modelling of how people's opinions evolve over time. This talk will focus on 'bounded confidence' models, in which people only take into account the opinion of others if they are sufficiently close in 'opinion space' (i.e. they already somewhat agree on a topic).

I will first introduce agent-based, ODE, SDE, and PDE models for these dynamics, before focusing on the (nonlocal, nonlinear) PDE case, which has analogues with approaches used in statistical mechanics. The main focus of the talk will be on the complex dynamics that arise; the presence of 'phase transitions' under varying parameters in the model; the crucial choice of boundary conditions; and the introduction of 'radicals', whose opinion is invariant and act as an 'external potential'.

Joint work with Beth Gooding, Greg Pavliotis, and Hannah Short

Week 8. Kerstin Lux (TU Munich) - Uncertainty Quantification of Bifurcations in Random Ordinary Differential Equations

(joint with Christian Kuehn)

Subsystems of the earth might undergo critical transitions under sustained global warming with severe impacts on various ecosystems and human habitat. This phenomenon is not restricted to climate science but appears also in ecology and epidemiology [1].
Here, we approach these critical transitions mathematically in terms of bifurcation theory for nonlinear ordinary differential equations. It is well-known in the theory of dynamical systems that parameter variation can induce bifurcations. Our main question of interest is how uncertainties in system parameters propagate through the possibly highly nonlinear dynamical system and affect the occurrence of different bifurcation types [2].
While some types cause rather smooth qualitative changes, others can cause the system to suddenly jump to a new equilibrium far away from previous dynamics. In this talk, I will combine known statistical and probabilistic concepts with bifurcation theory to contribute to a risk assessment of the exposure to critical transitions.
[1] Christian Kuehn and Christian Bick. A universal route to explosive phenomena. Science Advances, 7(16), 2021.
[2] Christian Kuehn and Kerstin Lux. Uncertainty Quantification of Bifurcations in Random Ordinary Differential Equations. SIAM J. Appl. Dyn. Syst., 20(4):2295-2334, 2021.

Week 9. Thomasina Ball (Warwick) - Blister dynamics

Many problems involve the spreading of a viscous fluid underneath a surface skin or crust, such as the intrusion of magma into the crust, the formation of ordered wrinkle patterns for the production of microfluidic devices, and the reopening of airways in biological fluid mechanics. A characteristic of these types of problems is that the spreading is controlled by the physics at the fluid front rather than a bulk similarity solution due to the singular nature of the contact line. This leads to a matching problem between a quasi-static interior and the behaviour of the peeling region at the front.Typically these studies have considered the skin to be elastic, however in many cases a more viscous or plastic description might be a more relevant model. In this talk, I will first consider viscous flow underneath an elastic sheet and the choice of regularisation at the front to deal with the singular behaviour there. I will then characterise the governing equations for the deformation of a viscoplastic plate (a Herschel-Bulkley rheology) due to a uniform load. Finally, I will touch upon my current research looking at the deformation of a viscoplastic plate due to viscous flow underneath.

Week 10. Matthew Colbrook (Cambridge/Paris) - Koopman operators and the computation of spectral properties in infinite dimensions

Note: Special time at 11:00 am due to "Christmas lunch".

Koopman operators are infinite-dimensional operators that globally linearise nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. Their increasing popularity, dubbed "Koopmanism", includes 10,000s of articles over the last decade. However, Koopman operators can have continuous spectra and lack finite-dimensional invariant subspaces, making computing their spectral properties a considerable challenge. This talk describes data-driven algorithms with rigorous convergence guarantees for computing spectral properties of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), the first scheme for computing the spectra and pseudospectra of general Koopman operators from trajectory data without spectral pollution. By combining ResDMD and the resolvent, we compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. When computing the continuous and discrete spectrum, explicit convergence theorems provide high-order convergence, even for chaotic systems. Kernelized variants of our algorithms allow for dynamical systems with a high-dimensional state-space. These methods are placed within a broader infinite-dimensional numerical linear algebra programme on infinite-dimensional spectral computations. We end with computing the spectral measures of a protein molecule (20,046-dimensional state-space) and computing nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number greater than 10^5 (295,122-dimensional state-space).

Term 1

Week 1. Cicely Macnamara (Glasgow) - An agent-based model of the tumour microenvironment

The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modelling. In this talk I shall present a 3D individual-based force-based model for tumour growth and development in which we simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent is fully realised, for example, cells are described as viscoelastic sphere with radius and centre given within the off-lattice model. Interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells, as well as intercellular aspects such as cell phenotypes.

Week 2. Nabil Fadai (Nottingham) - Semi-infinite travelling waves arising in moving-boundary reaction-diffusion equations

Travelling waves arise in a wide variety of biological applications, from the healing of wounds to the migration of populations. Such biological phenomena are often modelled mathematically via reaction-diffusion equations; however, the resulting travelling wave fronts often lack the key feature of a sharp ‘edge’. In this talk, we will examine how the incorporation of a moving boundary condition in reaction-diffusion models gives rise to a variety of sharp-fronted travelling waves for a range of wavespeeds. In particular, we will consider common reaction-diffusion models arising in biology and explore the key qualitative features of the resulting travelling wave fronts.

Week 3. Jemima Tabeart (Edinburgh) - Parallelisable preconditioners for saddle point weak-constraint 4D-Var

The saddle point formulation of weak-constraint 4D-Var offers the possibility of exploiting modern computer architectures. Developing good preconditioners which retain the highly-parallelisable nature of the saddle point system has been an area of recent research interest, especially for applications to numerical weather prediction. In this presentation I will present new proposals for preconditioners for the model and observation error covariance terms which explicitly incorporate model information and correlated observation error information respectively for the first time. I will present theoretical results comparing our new preconditioners to existing standard choices of preconditioners. Finally I will present two numerical experiments for the heat equation and Lorenz 96 and show that even when our theoretical assumptions are not completely satisfied, our new preconditioners lead to improvements in a number of settings.

Week 4. Jamie Foster (Portsmouth) - Mathematical modeling of brewing espresso

We give a brief introduction to current practices in cafe-style espresso extraction. Making the tastiest cup is considered an art rather than a science, however, processes can be made more systematic by better understanding the physical processes underlying brewing. We develop a physics-based mathematical model for the extraction process. Owing to the disparity in lengthscales between that of a grain within the "puck" and the depth of the whole "puck", the model can be systematically reduced via asymptotic homogenisation. The reduced model can then be solved by numerical methods. We will show that the model is able to reproduce some of the trends that are observed in experimental data and in practice. We also discuss possible strategies to make espresso more efficiently and more reproducibly.

Week 5. Louise Dyson (Warwick) - From ants to COVID-19: applications of mathematical modelling in biology and epidemiology

Like many applied mathematicians, I am interested in a variety of ways mathematical modelling may be used to understand the world. In this talk I will discuss two applications. Firstly investigating the phenomenon of noise-induced bistable states, inspired by the organisation of ant colonies, and extending this to consider how coupling together multiple bistable systems may induce synchronisation. Secondly describing my recent work during the COVID-19 pandemic analysing and modelling variants of concern.

Week 6. Upanshu Sharma (Berlin) - Variational structures beyond gradient flows

Gradient flows is an important subclass of evolution equations, whose solution dissipates an energy ‘as fast as possible’. This distinguishing feature endows these equations with a natural variational structure, which has received enormous attention over the last two decades. In recent years, it has become clear that if the gradient-flow equation originates from an underlying (reversible) stochastic particle system, then often the aforementioned variational structure is an exact decomposition of the large-deviation rate functional for the particle system. However, this decomposition and the corresponding gradient-flow structure breaks down if the underlying particles have additional non-dissipative effects (for instance in the case of non-reversible independent particles), even though the large-deviation rate functional is still available. Using the guiding example of independent non-reversible Markov jump particles, in this talk, I will discuss the various features of the rate functional and how it connects to relative entropy and Fisher information. Furthermore, I will show that if the underlying particle system is augmented with fluxes, then it is possible to derive gradient-flow-type structures in the non-reversible setting.

Week 7. Olga Mula (Paris Dauphine) - Optimal State and Parameter Estimation Algorithms and Applications to Biomedical Problems

In this talk, I will present an overview of recent works aiming at solving inverse problems (state and parameter estimation) by combining optimally measurement observations and parametrized PDE models. After defining a notion of optimal performance in terms of the smallest possible reconstruction error that any reconstruction algorithm can achieve, I will present practical numerical algorithms based on nonlinear reduced models for which we can prove that they can deliver a performance close to optimal. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic notions of accuracy quantification. I will illustrate the performance of the approach on simple benchmark examples and we will also discuss applications of the methodology to biomedical problems which are challenging due to shape variability.

Week 8. Clarice Poon (Bath) - Smooth bilevel programming for sparse regularization

Nonsmooth regularisers are widely used in machine learning for enforcing solution structures (such as the l1 norm for sparsity or the nuclear norm for low rank). State of the art solvers are typically first order methods or coordinate descent methods which handle nonsmoothness by careful smooth approximations and support pruning. In this work, we revisit the approach of iteratively reweighted least squares (IRLS) and show how a simple reparameterization coupled with a bilevel resolution leads to a smooth unconstrained problem. We are therefore able to exploit the machinery of smooth optimisation, such as BFGS, to obtain local superlinear convergence. The result is a highly versatile approach, handles both convex and non-convex regularisers, and different optimisation formulations (e.g. basis pursuit and lasso). We show that this is able to significantly outperform state of the art methods for a wide range of problems.

Week 9. Tom Montenegro-Johnson (Birmingham) - Soft, long, and thoroughly absorbent: unpublished works on looped active filaments and responsive hydrogels

Modern manufacturing techniques are now sufficiently advanced to create swimming microbots made of soft, programmable materials and catalytic fluid-driving components. This combination can endow microbots with a wealth of complex dynamical behaviours, arising from the multiphysical interactions of heterogenous materials, fluid domain, and swimmer geometry.

I will discuss 2 recent works in this field. In the first, we extend our recent slender phoretic theory to model and anbalyse chemically-active autophoretic slender loops, including a theme with variations on the torus, and a trefoil knot. In the second, we consider the swelling and shrinking dynamics of a thermoresponsive sphere of hydrogel. An effort will be made to relate the two in a wider context.

Week 10. Peter Baddoo (MIT) - Integrating physical laws into data-driven system identification

Incorporating partial knowledge of physical laws into data-driven architectures can improve the accuracy, generalisability, and robustness of the resulting models. In this work, we demonstrate how physical laws – such as symmetries, invariances, and conservation laws – can be integrated into the dynamic mode decomposition (DMD). DMD is a widely-used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD frequently produces models that are sensitive to noise, fail to generalise outside the training data, and violate basic physical laws. Our physics-informed DMD (piDMD) optimisation restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical properties – conservative, self-adjoint, local, causal, and shift-invariant – and derive closed-form solutions and efficient algorithms for the corresponding piDMD optimisations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of challenging problems in the physical sciences, including travelling-wave systems, the Schrödinger equation, solute advection-diffusion, and three-dimensional transitional channel flow. In each case, piDMD significantly outperforms standard DMD in metrics such as spectral identification, state prediction, and estimation of optimal forcings and responses. We also demonstrate how to model high-dimensional nonlinear structures via kernel regression.