AMCS Seminar Series

Transport and diffusion are essential in biological processes, generating molecular fluxes that sustain life. Classically, these fluxes are described by Fick’s laws for random walks near thermal equilibrium. However, active systems locally inject energy into their surrounding medium, driving non-equilibrium diffusion for which a general theory remains elusive. In this talk, I will discuss non-equilibrium processes two distinct types of biological active matter: active baths and active carpets. In the former, suspensions of active particles can enhance mixing and transport cargo using collective hydrodynamic entrainment. In the latter, active carpets, biological activity is highly concentrated on surfaces such as microbial biofilms. They can generate flows to attract nutrients from the bulk, by clustering and forming topological defects. Moreover, active carpets produce anisotropic and space-dependent diffusion, for which we derived generalized Fick’s laws. We solved two archetypal problems using these laws: First, considering sedimentation towards an active carpet, we find a self-cleaning effect where surface-driven fluctuations can repel particles. Second, considering diffusion from a source to an active sink, say nutrient capture by suspension feeders, we derived the enhanced molecular flux compared to thermal diffusion. Hence, our results could elucidate specific non-equilibrium properties of active coating materials and life at interfaces.

[1] Liu et al., arXiv preprint arXiv:2509.25068
[2] Jin et al., Physical Review Letters 127: 088006 (2021)
[3] Guzman-Lastra et al., Nature Communications 12: 1906 (2021)
[4] Mathijssen et al., Physical Review Letters 121: 248101 (2018)
[5] Mathijssen et al., Nature 571: 560 (2019) 

BIO: Arnold Mathijssen completed his physics undergraduate at University College London (2012), his PhD with Julia Yeomans FRS at the University Oxford (2016), and a postdoc in bioengineering with Manu Prakash at Stanford University (2020). He is now a faculty member at the University of Pennsylvania and director of the Penn Working Group on Environmental and Biological Fluid Dynamics. Arnold received the ‘30 under 30’ Award by Scientific American, the Charles Kittel Award by the American Physical Society, the Sir Sam Edwards PhD Thesis Prize by the UK Institute of Physics, the HFSP Cross-Disciplinary Fellowship, and the Paul Sniegowski Award for Mentorship of Undergraduate Research. Arnold is also a popular science communicator known for culinary fluid mechanics and the science of pour-over coffee, with appearances at Xfinity Live! and Mornings with Simi as well as coverage in The New York Times, CNN, The Guardian, FOX News, USA Today, Bean Scene, and Food & Wine Magazine.

During the last 20 years there has been a lot of progress in applying neural networks (NNs) to many machine learning tasks. However, their employment in scientific machine learning with the purpose of learning complex responses of physical systems from experimental measurements has been explored much less. In this talk, we will consider learning of heterogeneous material responses as an exemplar problem to investigate automated physical model discovery from experimental data. In particular, we propose to parameterize the mapping between excitation and corresponding system responses in the form of nonlocal neural operators, and infer the neural network parameters from experimental measurements. As such, the model is built as mappings between infinite-dimensional function spaces, and the learnt network parameters are resolution-agnostic. Moreover, the nonlocal operator architecture also allows the incorporation of fundamental mathematical and physics knowledge. Both properties improve the learning efficacy and robustness from scarse and noisy measurements.

To demonstrate the applicability of our nonlocal operator learning framework, two typical scenarios will be discussed: (1) learning of a material-specific constitutive law, (2) development of a foundation constitutive law across multiple materials. As an application, we learn material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and we will show that the learnt model substantially outperforms conventional constitutive models.

Hyperbolic equations are used extensively in applications including fluid dynamics, astrophysics, electro-magnetism, semi-conductor devices, and biological sciences.  High order accurate numerical methods are efficient for solving such partial differential equations, however they are difficult to design because solutions may contain discontinuities.

In this talk we will survey several types of high order numerical methods for such problems, including weighted essentially non-oscillatory (WENO) finite difference and finite volume methods, discontinuous Galerkin finite element methods, and spectral methods.  We will discuss essential ingredients, properties and relative advantages of each method, and provide comparisons among these methods.  Recent development and applications of these methods will also be discussed.

Is there an algorithm that learns the best fit parameters of a Transformer to any dataset?  If I trained a neural sequence model and promised you it is equivalent to a program, how would you even be convinced?  Modern RNN’s are functions that admit parallelizable recurrence; what is the design space of parallelizable recurrences?  Are there unexplored function families that lie between RNN’s and Transformers?  We explore these questions from first principles starting with state, polynomials, and parallelism.

In this talk, I begin with a review of different geometric flows (PDEs) including mean curvature (curve shortening) flow, surface diffusion flow, Willmore flow, Gauss curvature flow, etc., which arise from materials science, interface dynamics in multi-phase flows, biology membrane, computer graphics, geometry, etc. Different mathematical formulations and numerical methods for mean curvature flow are then discussed. In particular, an energy-stable linearly implicit parametric finite element method (PFEM) is presented in details. Then the PFEM is extended to surface diffusion flow and anisotropic surface diffusion flow, and a structure-preserving implicit PFEM is proposed. Finally, sharp interface models and their PFEM approximations are presented for solid-state dewetting. This talk is based on joint works with Harald Garcke, Wei Jiang, Yifei Li, Robert Nuernberg, Tiezheng Qian, David Srolovitz, Dongmin Wang, Yan Wang and Quan Zhao.

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We take a variational approach to understand hydrodynamic limit of (global) action minimizing Lagrangian collective dynamics, of weakly interacting deterministic particles.

We convert the problem into one of studying multi-scale limit theorems on Hamilton-Jacobi equation in space of probability measures. We derive an effective Hamiltonian and its associated variational problem.  We make extensive use of recent theories on optimal transport, first order analysis in Alexandrov metric spaces (for understanding the PDEs involved), and the weak KAM theory in finite dimensions (for the averaging step which implicitly connects with micro-canonical ensemble type arguments).

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