Fast-slow systems evolve simultaneously on two disparate time-scales. The fast evolution severely impedes the runtime of simulating such systems. In applications, however, it often suffices to obtain knowledge about the slow evolution only. Homogenisation in time can be used to derive a system that describes the slow evolution to leading-order. For a simple fast-slow system, we will discuss an extension of the homogenisation procedure that allows to derive a system that describes the slow evolution up to second-order. Additionally, we will discuss how the slow leading- and second-order evolution can be interpreted in a thermodynamic sense.
Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816
Mathematics has always been inspired by thinking about the natural world and aiming to understand phenomena surrounding us. The talk is aimed to be an invitation to follow on a naturally personal route, exploring applications of mathematical research in the eld of natural sciences. As for the rst part, we will depart from a naively posed question in image processing and consider it as leading to work on a particular class of kernels for data-driven analysis on spheres. Moving beyond the original problem, insights into classes of zonal kernels for the sphere also provide a better understanding of widely used models of similarity in the Euclidean space. To move the other way round, we will consider a completely dierent eld of application, asking what mathematical tools can be applied to gain a better understanding of complex adaptive systems. Taking in the abstract perspective of mathematics, allows for a generic view onto such systems, leading to an approach for inference, which is independent of the concrete instantiation of the system. Doing so, we can discuss more general questions on what constitutes the essence of evolution and how to identify its traces in natural phenomena.
Summarizing, both examples stand for the strong belief in interdisciplinary scientic work as a joint adventure at eye level, rather than the softening blur of a little bit of everything.
Hybrid-Veranstaltung: Präsenzvortrag im Hörsaal A027 und gleichzeitige Übertragung per Zoom mit folgendem Link:
Join Zoom Meeting: https://lmu-munich.zoom.us/j/99946902916?pwd=UWM5SGtIL091NmdjU3BHVVpOU0lEdz09
Meeting ID: 999 4690 2916, Passcode: 695211
Neural ordinary differential equations describe how values change in time with neural networks. This is the reason why they gained importance in modeling sequential data, especially when the observations are made at irregular intervals. We will discuss an alternative approach that directly models the solution curves - the flow of an ODE - with a neural network. This immediately eliminates the need for expensive numerical solvers while still maintaining the modeling capability of neural ODEs. We will show several flow architectures suitable for different applications by establishing precise conditions on when a function defines a valid flow. The models provide, apart from computational efficiency, an empirical evidence of favorable generalization performance via applications in time series modeling, forecasting, and density estimation.
TBA
Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816
Graph Convolutional Networks (GCNs) have emerged as powerful tools for learning on network structured data. Although empirically successful, GCNs exhibit certain behaviour that has no rigorous explanation -- for instance, the performance of GCNs significantly degrades with increasing network depth, whereas it improves marginally with depth using skip connections. This paper focuses on semi-supervised learning on graphs, and explains the above observations through the lens of Neural Tangent Kernels (NTKs). We derive NTKs corresponding to infinitely wide GCNs (with and without skip connections). Subsequently, we use the derived NTKs to identify that, with suitable normalisation, network depth does not always drastically reduce the performance of GCNs -- a fact that we also validate through extensive simulation. Furthermore, we propose NTK as an efficient `surrogate model' for GCNs that does not suffer from performance fluctuations due to hyper-parameter tuning since it is a hyper-parameter free deterministic kernel. The efficacy of this idea is demonstrated through a comparison of different skip connections for GCNs using the surrogate NTKs.
I this talk I will introduce a class of discrete statistical models to represent context-specific conditional independence relations for discrete data. These models can also be represented by sequences of context-DAGs (directed acyclic graphs). We prove that two of these models are statistically equivalent if and only their contexts are equal and the context DAGs have the same skeleton and v-structures. This is a generalization of the Verma and Pearl criterion for equivalence for DAGs. This is joint work with Liam Solus. A 3 minute video abstract for this talk is available at https://youtu.be/CccVNRFmR1I .
One major question in theoretical biology and economics is why unrelated people would help each other (i.e., why they would pay a cost to provide a benefit to someone else). Direct reciprocity provides a possible answer: we might help others today because this might increase the chance they help us tomorrow. This logic is typically formalized with a game called the repeated prisoner's dilemma. The talk consists of two parts. The first part will provide some background on previous models of direct reciprocity, and the kinds of dynamical systems they give rise to. In the second part I will discuss an axiomatic approach to construct cooperative and stable strategies of the repeated prisoner's dilemma. These strategies do not depend on the entire past history of the game, but only on the past n joint interactions.
In this talk the relationship between strongly chordal graphs and m-saturated vines (regular vines with certain nodes removed or assigned with independence copula) is proved. Moreover, an algorithm to construct an m-saturated vine structure corresponding to a strongly chordal graph is provided. When the underlying data is sparse this approach leads to improvements in an estimation process as compared to current heuristic methods. Furthermore due to reduction of model complexity it is possible to evaluate all vine structures as well as to fit non-simplified vines.