A Finite Expression Method for Solving High-Dimensional Committor Problems (2306.12268v2)
Abstract: Transition path theory (TPT) is a mathematical framework for quantifying rare transition events between a pair of selected metastable states $A$ and $B$. Central to TPT is the committor function, which describes the probability to hit the metastable state $B$ prior to $A$ from any given starting point of the phase space. Once the committor is computed, the transition channels and the transition rate can be readily found. The committor is the solution to the backward Kolmogorov equation with appropriate boundary conditions. However, solving it is a challenging task in high dimensions due to the need to mesh a whole region of the ambient space. In this work, we explore the finite expression method (FEX, Liang and Yang (2022)) as a tool for computing the committor. FEX approximates the committor by an algebraic expression involving a fixed finite number of nonlinear functions and binary arithmetic operations. The optimal nonlinear functions, the binary operations, and the numerical coefficients in the expression template are found via reinforcement learning. The FEX-based committor solver is tested on several high-dimensional benchmark problems. It gives comparable or better results than neural network-based solvers. Most importantly, FEX is capable of correctly identifying the algebraic structure of the solution which allows one to reduce the committor problem to a low-dimensional one and find the committor with any desired accuracy.
- Mordecai Avriel. Nonlinear programming: analysis and methods. Courier Corporation, 2003.
- Diffusion maps tailored to arbitrary non-degenerate itô processes. Applied and Computational Harmonic Analysis, 48(1):242–265, 2020.
- Protein conformational transitions: the closure mechanism of a kinase explored by atomistic simulations. Journal of the American Chemical Society, 131(1):244–250, 2009.
- Friedrichs learning: Weak solutions of partial differential equations via deep learning. arXiv preprint arXiv:2012.08023, 2020.
- Committor functions via tensor networks. Journal of Computational Physics, 472:111646, 2023.
- Neural-network-based Approximations for Solving Partial Differential Equations. Comm. Numer. Methods Engrg., 10:195–201, 1994.
- Towards a theory of transition paths. Journal of statistical physics, 123(3):503–523, 2006.
- Transition-path theory and path-finding algorithms for the study of rare events. Annual review of physical chemistry, 61:391–420, 2010.
- The deep ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat., 6:1–12, 2018.
- Computing committors via mahalanobis diffusion maps with enhanced sampling data. The Journal of Chemical Physics, 157:214107, 2022.
- Computing committors in collective variables via mahalanobis diffusion maps. Applied and Computational Harmonic Analysis, 64:62–101, 2023.
- Roger Fletcher. Practical methods of optimization. John Wiley & Sons, 2013.
- Numerical methods for special functions. Society for Industrial and Applied Mathematics, 3600 University City Science Center Philadelphia, PA, United States, 2007.
- Large conformational changes in proteins: signaling and other functions. Current opinion in structural biology, 20(2):142–147, 2010.
- Bag of tricks for image classification with convolutional neural networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 558–567, 2019.
- Solving for high-dimensional committor functions using artificial neural networks. Research in the Mathematical Sciences, 6(1):1–13, 2019.
- Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
- Artificial neural networks for solving ordinary and partial differential equations. IEEE transactions on neural networks, 9(5):987–1000, 1998.
- Point cloud discretization of fokker–planck operators for committor functions. Multiscale Modeling & Simulation, 16(2):710–726, 2018.
- Partial differential equations with numerical methods, volume 45. Springer, 2003.
- Computing committor functions for the study of rare events using deep learning. The Journal of Chemical Physics, 151:054112, 2019.
- Finite expression method for solving high-dimensional partial differential equations. arXiv preprint arXiv:2206.10121, 2022.
- Transition-state optimization on free energy surface: Toward solution chemical reaction ergodography. International journal of quantum chemistry, 70(1):95–103, 1998.
- Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. In International Conference on Learning Representations, 2021.
- Learning with rare data: Using active importance sampling to optimize objectives dominated by rare events, 2020.
- Learning representations by back-propagating errors. nature, 323(6088):533–536, 1986.
- Dgm: A deep learning algorithm for solving partial differential equations. Journal of computational physics, 375:1339–1364, 2018.
- Reinforcement learning: An introduction. MIT press, 2018.
- Lloyd N Trefethen. Spectral methods in MATLAB. SIAM, 2000.
- Deep nitsche method: Deep ritz method with essential boundary conditions. Communications in Computational Physics, 29(5):1365–1384, 2021.
- Nucleation and growth in pressure-induced phase transitions from molecular dynamics simulations: Mechanism of the reconstructive transformation of nacl to the cscl-type structure. Physical review letters, 92(25):250201, 2004.
- Weak adversarial networks for high-dimensional partial differential equations. Journal of Computational Physics, 411:109409, 2020.
- Phase transitions and nucleation mechanisms in metals studied by nanocalorimetry: A review. Thermochimica Acta, 603:2–23, 2015.