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Molecular dynamics simulations have the potential to provide atomic-level detail and insight to important questions in chemical physics that cannot be observed in typical experiments. However, simply generating a long trajectory is…
This paper describes work carried out on a model for the evolution of graph classes in complex objects. By defining evolution rules and propagation strategies on graph classes, we aim to define a user-definable means to manage data…
Recent research has extended methods from the fields of thermodynamics and statistical mechanics into other disciplines. Most notably, one recent work creates a unified theoretical framework to understand evolutionary biology, machine…
Signal processing traditionally relies on classical statistical modeling techniques. Such model-based methods utilize mathematical formulations that represent the underlying physics, prior information and additional domain knowledge. Simple…
Although deep models achieve high predictive performance, it is difficult for humans to understand the predictions they made. Explainability is important for real-world applications to justify their reliability. Many example-based…
Deep learning models have gained great popularity in statistical modeling because they lead to very competitive regression models, often outperforming classical statistical models such as generalized linear models. The disadvantage of deep…
The basic mechanics of evolution have been understood since Darwin. But debate continues over whether macroevolutionary phenomena are driven primary by the fitness structure of genotype space or by ecological interaction. In this paper we…
Recent studies of in vitro evolution of DNA via protein binding indicate that the evolution behavior is qualitatively different in different parameter regimes. I here present a general theory that is valid for a wide range of parameters,…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
As metabolomics datasets are becoming larger and more complex, there is an increasing need for model-based data integration and analysis to optimally leverage these data. Dynamical models of metabolism allow for the integration of…
Scientists investigate the dynamics of complex systems with quantitative models, employing them to synthesize knowledge, to explain observations, and to forecast future system behavior. Complete specification of systems is impossible, so…
Real-world generalization, e.g., deciding to approach a never-seen-before animal, relies on contextual information as well as previous experiences. Such a seemingly easy behavioral choice requires the interplay of multiple neural…
Time evolution is formulated and discussed in the framework of Schroeder's functional equation. The proposed method yields smooth, continuous dynamics without the prior need for local propagation equations.
Molecular traits, such as gene expression levels or protein binding affinities, are increasingly accessible to quantitative measurement by modern high-throughput techniques. Such traits measure molecular functions and, from an evolutionary…
It is widely believed that engineering a model to be invariant/equivariant improves generalisation. Despite the growing popularity of this approach, a precise characterisation of the generalisation benefit is lacking. By considering the…
While there are many applications of ML to scientific problems that look promising, visuals can be deceiving. Using numerical analysis techniques, we rigorously quantify the accuracy, convergence rates, and generalization bounds of certain…
We propose generalized Fermat's conjecture in the framework of arithmetic dynamics, and give evidences. The multi-indexed version is added.
Here we show that a particular one-parameter generalization of the exponential function is suitable to unify most of the popular one-species discrete population dynamics models into a simple formula. A physical interpretation is given to…
Data-driven methods for the identification of the governing equations of dynamical systems or the computation of reduced surrogate models play an increasingly important role in many application areas such as physics, chemistry, biology, and…