Data-driven Lie point symmetry detection for continuous dynamical systems

Symmetry detection, the task of discovering the underlying symmetries of a given dataset, has been gaining popularity in the machine learning community, particularly in science and engineering applications. Most previous works focus on detecting ‘canonical’ symmetries such as translation, scaling, and rotation, and cast the task as a modeling problem involving complex inductive biases and architecture design of neural networks. We challenge these assumptions and propose that instead of constructing biases, we can learn to detect symmetries from raw data without prior knowledge. The approach presented in this paper provides a flexible way to scale up the detection procedure to non-canonical symmetries, and has the potential to detect both known and unknown symmetries alike. Concretely, we focus on predicting the generators of Lie point symmetries of partial differential equations, more specifically, evolutionary equations for ease of data generation. Our results demonstrate that well-established neural network architectures are capable of recognizing symmetry generators, even in unseen dynamical systems. These findings have the potential to make non-canonical symmetries more accessible to applications, including model selection, sparse identification, and data interpretability

Information Decomposition Diagrams Applied beyond Shannon Entropy: A Generalization of Hu's Theorem

In information theory, one major goal is to find useful functions that summarize the amount of information contained in the interaction of several random variables. Specifically, one can ask how the classical Shannon entropy, mutual information, and higher interaction information functions relate to each other. This is formally answered by Hu's theorem, which is widely known in the form of information diagrams: it relates disjoint unions of shapes in a Venn diagram to summation rules of information functions; this establishes a bridge from set theory to information theory. While a proof of this theorem is known, to date it was not analyzed in detail in what generality it could be established. In this work, we view random variables together with the joint operation as a monoid that acts by conditioning on information functions, and entropy as the unique function satisfying the chain rule of information. This allows us to abstract away from Shannon's theory and to prove a generalization of Hu's theorem, which applies to Shannon entropy of countably infinite discrete random variables, Kolmogorov complexity, Tsallis entropy, (Tsallis) Kullback-Leibler Divergence, cross-entropy, submodular information functions, and the generalization error in machine learning. Our result implies for Chaitin's prefix-free Kolmogorov complexity that the higher-order interaction complexities of all degrees are in expectation close to Shannon interaction information. For well-behaved probability distributions on increasing sequence lengths, this shows that asymptotically, the per-bit expected interaction complexity and information coincide, thus showing a strong bridge between algorithmic and classical information theory

Glucocorticoid sensitivity in Behcet's disease

WOS: 000209773300007PubMed ID: 23781311Objective: Glucocorticoid (GC) sensitivity is highly variable among individuals and has been associated with susceptibility to develop (auto-) inflammatory disorders. The purpose of the study was to assess GC sensitivity in Behcet's disease (BD) by studying the distribution of four GC receptor (GR) gene polymorphisms and by measuring in vitro cellular GC sensitivity. Methods: Healthy controls and patients with BD in three independent cohorts were genotyped for four functional GR gene polymorphisms. To gain insight into functional differences in in vitro GC sensitivity, 19 patients with BD were studied using two bioassays and a whole-cell dexamethasone-binding assay. Finally, mRNA expression levels of GR splice variants (GR-alpha and GR-beta) were measured. Results: Healthy controls and BD patients in the three separate cohorts had similar distributions of the four GR polymorphisms. The Bcll and 9 beta minor alleles frequency differed significantly between Caucasians and Mideast and Turkish individuals. At the functional level, a decreased in vitro cellular GC sensitivity was observed. GR number in peripheral blood mononuclear cells was higher in BD compared with controls. The ratio of GR-alpha/GR-beta mRNA expression levels was significantly lower in BD. Conclusions: Polymorphisms in the GR gene are not associated with susceptibility to BD. However, in vitro cellular GC sensitivity is decreased in BD, possibly mediated by a relative higher expression of the dominant negative GR-b splice variant. This decreased in vitro GC sensitivity might play an as yet unidentified role in the pathophysiology of BD.The Dutch Arthritis AssociationThis work was supported by a grant from The Dutch Arthritis Association