相关论文: Neural Network-based High-index Saddle Dynamics Me…
The high-index saddle dynamics (HiSD) method [J. Yin, L. Zhang, and P. Zhang, {\it SIAM J. Sci. Comput., }41 (2019), pp.A3576-A3595] serves as an efficient tool for computing index-$k$ saddle points and constructing solution landscapes.…
We propose the nullspace-preserving high-index saddle dynamics (NPHiSD) method for degenerating multiple solution systems in constrained and unconstrained settings. The NPHiSD efficiently locates high-index saddle points and provides parent…
We introduce a generalized numerical algorithm to construct the solution landscape, which is a pathway map consisting of all stationary points and their connections. Based on the high-index optimization-based shrinking dimer (HiOSD) method…
High-index saddle dynamics (HiSD) is an effective approach for computing saddle points of a prescribed Morse index and constructing solution landscapes for complex nonlinear systems. However, for problems with ill-conditioned Hessians…
We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow…
The high-index saddle dynamics (HiSD) method provides a powerful framework for finding saddle points and constructing solution landscapes. While originally derived for nondegenerate critical points, HiSD has demonstrated empirical success…
We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-$k$ saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework,…
Saddle points provide a hierarchical view of the energy landscape, revealing transition pathways and interconnected basins of attraction, and offering insight into the global structure, metastability, and possible collective mechanisms of…
High-index saddle dynamics (HiSD) serves as a competitive instrument in searching the any-index saddle points and constructing the solution landscape of complex systems. The Lagrangian multiplier terms in HiSD ensure the Stiefel manifold…
We propose a model-free shrinking-dimer saddle dynamics for finding any-index saddle points and constructing the solution landscapes, in which the force in the standard saddle dynamics is replaced by a surrogate model trained by the Gassian…
This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting,…
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…
Saddle points play important roles as the transition states of activated process in gradient system driven by energy functional. However, for the same energy functional, the saddle points, as well as other stationary points, are different…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
Training deep neural network is a high dimensional and a highly non-convex optimization problem. Stochastic gradient descent (SGD) algorithm and it's variations are the current state-of-the-art solvers for this task. However, due to…
In this work, we propose Natural Hypergradient Descent (NHGD), a new method for solving bilevel optimization problems. To address the computational bottleneck in hypergradient estimation--namely, the need to compute or approximate Hessian…
We analyze the semi-implicit scheme of high-index saddle dynamics, which provides a powerful numerical method for finding the any-index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle…
Saddle dynamics is a time continuous dynamics to efficiently compute the any-index saddle points and construct the solution landscape. In practice, the saddle dynamics needs to be discretized for numerical computations, while the…
This paper proposes and analyzes an iterative minimization formulation for search- ing index-1 saddle points of an energy function. This formulation differs from other eigenvector-following methods by constructing a new objective function…