BACKGROUND: Protein structure prediction is one of the most important scientific problems, on the one hand, it is one of the NP-hard problems, and on the other hand, it has a wide range of applications including drug discovery and biotechnology development. Since experimental methods for structure determination remain expensive and time-consuming, computational structure prediction offers a scalable and cost-effective alternative and application of machine learning in structural biology has revolutionized protein structure prediction. Despite their success, machine learning methods face fundamental limitations in optimizing complex high-dimensional energy landscapes, which motivates research into new methods to improve the robustness and performance of optimization algorithms. RESULTS: This study presents a novel approach to protein structure prediction by integrating the Landscape Modification (LM) method with the Adam optimizer for OpenFold. The main idea is to change the optimization dynamics by introducing a gradient scaling mechanism based on energy landscape transformations. LM dynamically adjusts gradients using a threshold parameter and a transformation function, thereby improving the optimizer's ability to avoid local minima, more efficiently traverse flat or rough landscape regions, and potentially converge faster to global or high-quality local optima. By integrating simulated annealing into the LM approach, we propose LM SA, a variant designed to improve convergence stability while facilitating more efficient exploration of complex landscapes. CONCLUSION: We compare the performance of standard Adam, LM, and LM SA on different datasets and computational conditions. Performance was evaluated using Loss function values, predicted Local Distance Difference Test (pLDDT), distance-based Root Mean Square Deviation (dRMSD), and Template Modeling (TM) scores. Our results show that LM and LM SA outperform the standard Adam across all metrics, showing faster convergence and better generalization, particularly on proteins not included in the training set. These results demonstrate that integrating landscape-aware gradient scaling into first-order optimizers advances research in computational optimization and improves prediction performance for complex problems such as protein folding.

Given a target distribution π and an arbitrary Markov infinitesimal generatorLon a finite state space χ, we develop three structured and inter-related approaches to generate new reversiblizations fromL. The first approach hinges on a geometric perspective, in which we view reversiblizations as projections onto the space of π-reversible generators under suitable information divergences such asf-divergences. With different choices of functionsf, we not only recover nearly all established reversiblizations but also unravel and generate new reversiblizations. Along the way, we unveil interesting geometric results such as bisection properties, Pythagorean identities, parallelogram laws and a Markov chain counterpart of the arithmetic-geometric-harmonic mean inequality governing these reversiblizations. This further serves as motivation for introducing the notion of information centroids of a sequence of Markov chains and to give conditions for their existence and uniqueness. Building upon the first approach, we view reversiblizations as generalized means. In this second approach, we construct new reversiblizations via different natural notions of generalized means such as the Cauchy mean or the dual mean. In the third approach, we combine the recently introduced locally-balanced Markov processes framework and the notion of convex ∗-conjugate in the study off-divergence. The latter offers a rich source of balancing functions to generate new reversiblizations.