Maria K. Cameron

University of Maryland, Department of Mathematics

Summer 2023

Topic 3. Data-driven methods and model reduction for the study of rare events in stochastic systems.

Prerequisites: linear algebra, multivariable calculus, elementary probability, and some programming experience.

Many physical processes such as conformal changes in biomolecules and switches between stable modes in nonlinear oscillators are modeled using stochastic differential equations with small noise. The events of interest, the transitions between metastable states of such systems, happen rarely on the timescale of the system. As a result, their study by means of direct simulations requires extremely large runtimes. We will explore alternative approaches to the study of rare events in such systems based on model reduction and various tools originating from data science and machine learning. For example, a reduced model for an array of nonlinear oscillators with periodic forcing and small noise can be a discrete-time Markov chain, whose stochastic matrix needs to be learned from data. A reduced model for a molecular system can be learned from data by means of training a neural network to represent the original high-dimensional model in a low-dimensional space. Such a neural network is called an autoencoder.

Kick-off. Project presentation. Slides.

Tutorials.

Room: MATH0411. Time: 10 AM -- 12 PM.

• June 13: Basics of probability theory. Markov Chains. Maria Cameron
  Probability.pdf
  
• June 14: Markov Chains. Maria Cameron
MarkovChains.pdf
• June 15: Stochastic differential equations. Maria Cameron
SDEs.pdf
• June 16: Transition path theory. Collective variables. Reduced dynamics. Approximation Theory. Neural networks. Autoencoders. Maria Cameron
  TPT.pdf NeuralNetworks4PDEs.pdf
  Additional reading on reduced dynamics, collective variables, and autoencoders:
  Wei Zhang, Carsten Hartmann, Christof Schuette, Effective dynamics along given reaction coordinates, and reaction rate theory(2016)
  Christoph Wehmeyer and Frank Noe, Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics(2017)
  Wei Chen, Hythem Sidky, and Andrew L. Ferguson, Capabilities and Limitations of Time-lagged Autoencoders for Slow Mode Discovery in Dynamical Systems (2019)


• June 20: The study of transitions in nonlinear oscillators with periodic forcing by means of Markov chains. Christopher Moakler
  NonLinearOscillatorTutorial.pdf
• June 21: PCA, LDA, Diffusion Maps, effective dimension, approximating the generator of a stochastic process. Shashank Sule
  DimensionalityReduction.pdf
• June 22: Functional analysis. Luis Suarez
  Functional_Analysis.pdf
• June 23: Random graph theory. Perrin Ruth
  GraphDataAnalysis.pdf

Lab projects for weeks 1 and 2.

Each student should do four or more labs on their choice, as many as time allows.

• Generating stochastic trajectories.
  Test cases: Euler-Maruyama for the overdamped Langevin dynamics with Mueller’s potential or the face potential.
  Mueller's potential is a popular 2D test problem. See e.g. Section 6.1 in https://arxiv.org/abs/2305.17112.
  Ref.: D. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations
  The partitioned Euler-Maryuama for a noisy nonlinear oscillator.
  Ref.: D. Higham, Numerical simulation of a linear stochastic oscillator with additive noise
• Metadynamics method for enhanced sampling.
  Extraction of a delta-net.
  A clear and concise description of the basic metadynamics algorithm is found in Li, Lin, and Ren https://arxiv.org/abs/1906.06285 (see Section C.2).
  For delta-net, see Algorithm 3 in Appendix B in https://arxiv.org/pdf/2208.13772.pdf.
• Computing the mean force and the diffusion matrix for collective variables.
  Test case: toy example with a two-well potential from Legoll and Lelievre https://arxiv.org/abs/0906.4865 (see eq. (47) and Section 4).
  The decription of this computation is given in Maragliano et al. String Method in Collective Variables... and
  in Appendix A of https://arxiv.org/pdf/2108.08979.pdf (the main formulas for computing the mean force and the diffusion matrix are
  Eqs. (A-6) and (A-7), respectively).
• Finding stable periodic solutions and saddle cycles for nonlinear oscillators.
  Test case: https://arxiv.org/pdf/2305.19459.pdf (see eq. (7))
• Plot the bifurcation diagram for Lorenz’63 (see fig. 15 in https://arxiv.org/pdf/1809.09987.pdf ).
  It is sufficient to let the parameter rho increase from 1 to 30. Then also plot the diagram decreasing rho from 30 to 1.
• Autoencoders and time-lagged autoencoders.
  Test case: Legoll and Lelievre https://arxiv.org/abs/0906.4865 (see eq. (47) and Section 4).
  See the online tutorial: https://deeptime-ml.github.io/latest/notebooks/tae.html
  A code written by A. Luke Evans based on this tutorial: autoencoder_example.ipynb and dataset.npz, a dataset for it.
• Diffusion maps for the committor problem. Test cases: Mueller's potential and Face potential.
• Diffusion maps for manifold learning. Test case: the emoji set.
  Diffusion%20map%20on%20facedata.ipynb reads and visualizes emoji data
  FaceData.mat emoji data
  colors.mat one more data file
  MakeEmojiData.m generates emaji data files in Matlab
• Neural network-based committor solvers. Test case: Mueller’s potential.
  A Python code for solving the committor problem using a variational neural-network-based solver
  for the overdamped Langevin dynamics in Face potential Face_NN_deltanet.ipynb and training points for it FaceDeltaNet.csv.
• Implement LDA to find the collective variable separating all four minima for LJ7 in 2D.
  A description of LJ7 in 2D is found in Section 4.2 in https://arxiv.org/abs/2108.08979
  Description: final.pdf (see Project 1).
  Data files: LJ7FreeEnergy.csv and TrajectoryCV_data.csv.
• Implement graph analysis algorithms: breadth-first search and depth-first search.
  Test cases: Erdos-Renyi random graphs, a configurational model with a power-law degree distribution.
  Ref. Introduction to algorithms, Cormen Leiserson, Rivest, Stein.
• Simulate the SIR model on random graph models. Compare your results with theoretical predictions.

Tentative research projects

The studens can pick a project from this list or propose their own project.

• Analysis of a chaotic oscillator with noise using the Markov chain approach.
  Case study: Aragwal, Wang, Balachandran (2021) (see eq. (8)).
  The goal is to find the transition rate as a function of the noise coefficient.
• Analysis of a circular array of oscillators by means of Markov chains.
  Case study: Cilenti, Cameron, Balachandran (2022) (see eq. (1) and the caption for fig. (2).)
  Start with two coupled oscillators. The goal is to find the transition rate as a function of the noise coefficient.
• Analysis of LJ7 in 2D by means of various model reduction techniques (PCA, LDA, diffusion maps, autoencoders).
  The goal is to find a good set of collective variables and determine the transition rate.
  See Section 4.2 in https://arxiv.org/abs/2108.08979 and
  Section 6.3 in https://arxiv.org/abs/2305.17112.
• Apply and/or develop model reduction methods to learn the collective variables approximating the dynamics well.
  Useful details are elaborated by Shashank Sule: AutoencodesProjects.pdf.
  A reasonable test problem to start with is Legoll's and Lelievre's 2D test problem: https://arxiv.org/abs/0906.4865 (see eq. (47) and Section 4)
• Study the effect of noise on Gissinger’s model for Chaotic reversals. Estimate the transition rate.
• Effect of noise on population dynamics.
  Case study 1: A prey-predator model with a role reversal (Li, Liu, Wei (2022).
  Case study 2. Sanchez-Garduno, Miramontes, Marquez-Lago, Role reversal in prey-predator interactions (2014).
• Develop a graph model for the evolution of holes in for the process of fracture of a polymer network.

Design by Michelle Cameron