232 research outputs found
Deep learning Markov and Koopman models with physical constraints
The long-timescale behavior of complex dynamical systems can be described by
linear Markov or Koopman models in a suitable latent space. Recent variational
approaches allow the latent space representation and the linear dynamical model
to be optimized via unsupervised machine learning methods. Incorporation of
physical constraints such as time-reversibility or stochasticity into the
dynamical model has been established for a linear, but not for arbitrarily
nonlinear (deep learning) representations of the latent space. Here we develop
theory and methods for deep learning Markov and Koopman models that can bear
such physical constraints. We prove that the model is an universal approximator
for reversible Markov processes and that it can be optimized with either
maximum likelihood or the variational approach of Markov processes (VAMP). We
demonstrate that the model performs equally well for equilibrium and
systematically better for biased data compared to existing approaches, thus
providing a tool to study the long-timescale processes of dynamical systems
Discovering Causal Relations and Equations from Data
Physics is a field of science that has traditionally used the scientific
method to answer questions about why natural phenomena occur and to make
testable models that explain the phenomena. Discovering equations, laws and
principles that are invariant, robust and causal explanations of the world has
been fundamental in physical sciences throughout the centuries. Discoveries
emerge from observing the world and, when possible, performing interventional
studies in the system under study. With the advent of big data and the use of
data-driven methods, causal and equation discovery fields have grown and made
progress in computer science, physics, statistics, philosophy, and many applied
fields. All these domains are intertwined and can be used to discover causal
relations, physical laws, and equations from observational data. This paper
reviews the concepts, methods, and relevant works on causal and equation
discovery in the broad field of Physics and outlines the most important
challenges and promising future lines of research. We also provide a taxonomy
for observational causal and equation discovery, point out connections, and
showcase a complete set of case studies in Earth and climate sciences, fluid
dynamics and mechanics, and the neurosciences. This review demonstrates that
discovering fundamental laws and causal relations by observing natural
phenomena is being revolutionised with the efficient exploitation of
observational data, modern machine learning algorithms and the interaction with
domain knowledge. Exciting times are ahead with many challenges and
opportunities to improve our understanding of complex systems.Comment: 137 page
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