Physics-Informed Tree Search for High-Dimensional Computational Design

TL;DR

This work tackles high-dimensional, expensive black-box design optimization in physics-informed contexts by marrying Monte Carlo Tree Search with continuous-space adaptations. The framework introduces adaptive, direction-aware sampling via a Logistic Surrogate, depth-aware window scaling, and population-based tree batching to robustly explore rugged landscapes while respecting physical constraints. Across 23 benchmark functions and real-world tasks—ranging from crystal-structure prediction to interatomic-potential fitting and continuum design—the method achieves superior convergence, reduced sample complexity, and consistent feasibility, outperforming standard metaheuristics in many cases. By acting as a decision layer that couples simulations, surrogate evaluation, and constraint handling, the approach promises scalable, interpretable, and transferable improvements for computational design workflows in materials science and engineering.

Abstract

High-dimensional design spaces underpin a wide range of physics-based modeling and computational design tasks in science and engineering. These problems are commonly formulated as constrained black-box searches over rugged objective landscapes, where function evaluations are expensive, and gradients are unavailable or unreliable. Conventional global search engines and optimizers struggle in such settings due to the exponential scaling of design spaces, the presence of multiple local basins, and the absence of physical guidance in sampling. We present a physics-informed Monte Carlo Tree Search (MCTS) framework that extends policy-driven tree-based reinforcement concepts to continuous, high-dimensional scientific optimization. Our method integrates population-level decision trees with surrogate-guided directional sampling, reward shaping, and hierarchical switching between global exploration and local exploitation. These ingredients allow efficient traversal of non-convex, multimodal landscapes where physically meaningful optima are sparse. We benchmark our approach against standard global optimization baselines on a suite of canonical test functions, demonstrating superior or comparable performance in terms of convergence, robustness, and generalization. Beyond synthetic tests, we demonstrate physics-consistent applicability to (i) crystal structure optimization from clusters to bulk, (ii) fitting of classical interatomic potentials, and (iii) constrained engineering design problems. Across all cases, the method converges with high fidelity and evaluation efficiency while preserving physical constraints. Overall, our work establishes physics-informed tree search as a scalable and interpretable paradigm for computational design and high-dimensional scientific optimization, bridging discrete decision-making frameworks with continuous search in scientific design workflows.

Figures (6)

• Figure 1: Illustration of the breadth of Monte Carlo Tree Search (MCTS) applications, ranging from strategic decision-making in games to real-world optimization and scientific design tasks. Most existing uses operate in discrete action spaces.
• Figure 2: Computational design using decision tree-based MCTS for continuous search spaces. (a) Different stages of MCTS, starting from a root node as the search tree grows. (b) Transition from a discrete to a continuous action space, where infinitely many possible directions surround a node. (c) Scaling window schemes within a tree allow deeper branches to focus exploitation near promising regions. (d) A typical continuous search space illustrated by a 2D objective function. The tree typically starts from the highest objective region of the landscape and branches out through local minima. A population (batch) of trees is spawned across the continuous search space. The first stage involves global exploration; in subsequent stages, fewer trees are spawned from promising solutions for local optimization. The number of trees is progressively reduced as the search converges toward the global optimum.
• Figure 3: Performance Benchmarking of Our MCTS-Based Optimisation on Test Functions. (a–c) Convergence curves (logarithm of objective function f with nuber of evaluation) for 30 independent runs, along with the sampling distribution from the search that obtained the best solution. The three benchmark functions are: Unimodal (F4), Multimodal (F11), and Fixed Dimensional (F14), respectively. (d–f) Comparison of the performance of c-MCTS, WOA, and PSO in terms of the best solutions obtained and their standard deviations over 30 independent trials for the functions shown in (a–c).
• Figure 4: Application of our MCTS optimization algorithm in crystal structure optimization at multiple scales. (a) The primary workflow for utilizing the algorithm in a crystal structure prediction framework, comprising a physical constraint handling module, a generator that creates physically valid configurations and converts between structural and parametric representations, the optimization algorithm, and a property evaluator or calculator. (b) Application in predicting the global minimum of a gold (Au) nanocluster (0D material) consisting of 35 atoms (105-dimensional space), explored in terms of energy stability, and comparison of the final configuration with the reference in terms of the pairwise interatomic distance histogram. (c) Application in sheet-like 2D materials: structure-oriented search for a metastable polymorph of silicon (silicene), targeting configurations with specific structural attributes. The unit cell predicted by MCTS, as well as the prediction accuracy across different polymorphs, is shown. (d) Assessment of performance in optimizing the periodic bulk (3D material) ground-state structure of silicon (cubic diamond), starting from an amorphous, high-energy configuration using an energy-based objective search. The final obtained configuration, the reference configuration, and their structural comparison, in terms of the RDF, are shown for different bounds (ranges of the search space) (10% and 30%).
• Figure 5: Application of the MCTS algorithm for optimizing potential energy models. (a) Workflow illustrating the application of the MCTS-based optimization framework for developing potential energy models. The process includes four major components: (i) generation of a reference dataset from DFT or other high-fidelity methods, (ii) a calculator that maps the predicted model parameters to physical material properties, (iii) the MCTS optimization engine, and (iv) a local optimizer used for fine-tuning model parameters. (b,c) Parity plots comparing the predicted energies and forces with reference DFT data for aluminum (Al) nanoclusters, along with associated prediction errors on the test dataset. (d,e) Performance benchmarking of the developed model against standard Al potentials from the literature in terms of energy and force prediction errors.