Shuting Gu

Shuting Gu, Xiang Zhou

Among numerical methods for partial differential equations arising from steepest descent dynamics of energy functionals (e.g., Allen-Cahn and Cahn-Hilliard equations), the convex splitting method is well-known to maintain unconditional energy stability for a large time step size. In this work, we show how to use the convex splitting idea to find transition states, i.e., index-1 saddle points of the same energy functionals. Based on the iterative minimization formulation (IMF) for saddle points (SIAM J. Numer. Anal., vol. 53, p1786, 2015), we introduce the convex splitting method to minimize the auxiliary functional at each cycle of the IMF. We present a general principle of constructing convex splitting forms for these auxiliary functionals and show how to avoid solving nonlinear equations. The new numerical scheme based on the convex splitting method allows for large time step sizes. The new methods are tested for the one dimensional Ginzburg-Landau energy functional in the search of the Allen-Cahn or Cahn-Hilliard types of transition states. We provide the numerical results of transition states for the two dimensional Landau-Brazovskii energy functional for diblock copolymers.

Yi Liu, Shuting Gu

The Cahn--Hilliard equation is a fundamental model for describing phase separation phenomena in binary mixtures. Traditional numerical methods, such as finite difference and finite element methods, often incur substantial computational cost, particularly when computing steady-state solutions in high-dimensional settings. To address this challenge, we propose a deep learning-based framework, namely the Deep Ritz method, for computing steady states of the Cahn--Hilliard equation under periodic boundary conditions. An enhanced augmented Lagrangian formulation is incorporated to strictly enforce the mass conservation constraint, while separable Fourier feature mappings are employed to naturally encode periodicity and enhance the representation of nontrivial solution structures. The proposed method exhibits a notable dual capability: it not only achieves fast convergence to steady states but also effectively identifies multiple nontrivial solutions corresponding to different local minimizers of the energy functional. Extensive numerical experiments in one-, two-, and three-dimensional cases demonstrate that the method can successfully capture a rich variety of phase separation patterns, including droplet-type, lamellar, and tubular structures, highlighting its effectiveness and robustness in exploring complex high-dimensional energy landscapes.

Jiayue Han, Shuting Gu, Xiang Zhou

Rare transition events in meta-stable systems under noisy fluctuations are crucial for many non-equilibrium physical and chemical processes. In these processes, the primary contributions to reactive flux are predominantly near the transition pathways that connect two meta-stable states. Efficient computation of these paths is essential in computational chemistry. In this work, we examine the temperature-dependent maximum flux path, the minimum energy path, and the minimum action path at zero temperature. We propose the StringNET method for training these paths using variational formulations and deep learning techniques. Unlike traditional chain-of-state methods, StringNET directly parametrizes the paths through neural network functions, utilizing the arc-length parameter as the main input. The tasks of gradient descent and re-parametrization in the string method are unified into a single framework using loss functions to train deep neural networks. More importantly, the loss function for the maximum flux path is interpreted as a softmax approximation to the numerically challenging minimax problem of the minimum energy path. To compute the minimum energy path efficiently and robustly, we developed a pre-training strategy that includes the maximum flux path loss in the early training stage, significantly accelerating the computation of minimum energy and action paths. We demonstrate the superior performance of this method through various analytical and chemical examples, as well as the two- and four-dimensional Ginzburg-Landau functional energy.

Shuting Gu, Hongqiao Wang, Xiang Zhou

The saddle point (SP) calculation is a grand challenge for computationally intensive energy function in computational chemistry area, where the saddle point may represent the transition state (TS). The traditional methods need to evaluate the gradients of the energy function at a very large number of locations. To reduce the number of expensive computations of the true gradients, we propose an active learning framework consisting of a statistical surrogate model, Gaussian process regression (GPR) for the energy function, and a single-walker dynamics method, gentle accent dynamics (GAD), for the saddle-type transition states. SP is detected by the GAD applied to the GPR surrogate for the gradient vector and the Hessian matrix. Our key ingredient for efficiency improvements is an active learning method which sequentially designs the most informative locations and takes evaluations of the original model at these locations to train GPR. We formulate this active learning task as the optimal experimental design problem and propose a very efficient sample-based sub-optimal criterion to construct the optimal locations. We show that the new method significantly decreases the required number of energy or force evaluations of the original model.

Shuting Gu, Xiang Zhou

Here we present a multiscale method to calculate the saddle point associated with the effective dynamics arising from a stochastic system which couples slow deterministic drift and fast stochastic dynamics. This problem is motivated by the transition states on free energy surfaces in chemical physics. Our method is based on the gentlest ascent dynamics which couples the position variable and the direction variable and has the local convergence to saddle points. The dynamics of the direction vector is derived in terms of the covariance function with respective to the equilibrium distribution of the fast stochastic process. We apply the multiscale numerical methods to efficiently solve the obtained multiscale gentlest ascent dynamics, {and discuss the acceleration techniques based on the adaptive idea.} The examples of stochastic ordinary and partial differential equations are presented.