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Derivative-free tree optimization for complex systems

Ye Wei, Bo Peng, Ruiwen Xie, Yangtao Chen, Yu Qin, Peng Wen, Stefan Bauer, Po-Yen Tung

Abstract

A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional derivative-free optimization techniques often rely on strong assumptions about objective functions, thereby failing at optimizing non-convex systems beyond 100 dimensions. Here, we present a tree search method for derivative-free optimization that enables accelerated optimal design of high-dimensional complex systems. Specifically, we introduce stochastic tree expansion, dynamic upper confidence bound, and short-range backpropagation mechanism to evade local optimum, iteratively approximating the global optimum using machine learning models. This development effectively confronts the dimensionally challenging problems, achieving convergence to global optima across various benchmark functions up to 2,000 dimensions, surpassing the existing methods by 10- to 20-fold. Our method demonstrates wide applicability to a wide range of real-world complex systems spanning materials, physics, and biology, considerably outperforming state-of-the-art algorithms. This enables efficient autonomous knowledge discovery and facilitates self-driving virtual laboratories. Although we focus on problems within the realm of natural science, the advancements in optimization techniques achieved herein are applicable to a broader spectrum of challenges across all quantitative disciplines.

Derivative-free tree optimization for complex systems

Abstract

A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional derivative-free optimization techniques often rely on strong assumptions about objective functions, thereby failing at optimizing non-convex systems beyond 100 dimensions. Here, we present a tree search method for derivative-free optimization that enables accelerated optimal design of high-dimensional complex systems. Specifically, we introduce stochastic tree expansion, dynamic upper confidence bound, and short-range backpropagation mechanism to evade local optimum, iteratively approximating the global optimum using machine learning models. This development effectively confronts the dimensionally challenging problems, achieving convergence to global optima across various benchmark functions up to 2,000 dimensions, surpassing the existing methods by 10- to 20-fold. Our method demonstrates wide applicability to a wide range of real-world complex systems spanning materials, physics, and biology, considerably outperforming state-of-the-art algorithms. This enables efficient autonomous knowledge discovery and facilitates self-driving virtual laboratories. Although we focus on problems within the realm of natural science, the advancements in optimization techniques achieved herein are applicable to a broader spectrum of challenges across all quantitative disciplines.
Paper Structure (22 sections, 11 equations, 5 figures, 2 tables)

This paper contains 22 sections, 11 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Discussions
  4. Methods

Figures (5)

  • Figure 1: Principle of derivative-free stochastic tree search.(a) Feature engineering of complex systems. (b) Stochastic Expansion. (c) Conditional selection. (d) Stochastic rollout. (e) Dynamic confidence upper bound. (f and g) An autonomous virtual lab consisting of DOTS, neural network and simulators designed for various DFO applications in materials, instruments, physics, and biology.
  • Figure 2: (Caption next page.)
  • Figure 2: Benchmark study and algorithm analysis (second draft).(a,b,c) Benchmark studies on Rastrigin 1000d, Ackley 200d, Rosenbrock 100d respectively. (d) Convergence ratio on various functions of different dimensions. DOTS converges on all synthetic functions with more than $90\%$ ratio, whereas none of the other algorithms can converge at all (0$\%$). (e) DOTS explores a rugged ground-truth landscape using stochastic expansion and $DUCB$ inequality. During stochastic rollout, $DUCB$ of all leaf nodes compare to that of the root, DOTS stays with the root node till leaf node with lower $DUCB$ is found. (f) Top-visit sampling. DOTS iteratively samples data points from other local minima. (g) Ablation study on the individual mechanism. Each has been performed 10 times. (h) 2-dimensional U-MAP representation of the input distribution of DOTS and its ablated variants on Rosenbrock-100d. (i) DOTS overcomes the local minima in a flat landscape by using local backpropagation to create a local $DUCB$ gradient 'ladder'. (j) Adaptive exploration, which keeps the exploration weights at the same level as the overall ground-truth distribution. (k) Evolution path of DOTS in ground-truth space in Rosenbrock-100d. (l) Individual evolution path of DOTS and its ablated variants visualized in separated subplots from (h).
  • Figure 3: (Caption next page.)
  • Figure 3: Self-driving virtual laboratories for complex systems.(a) Optimizing the mechanical properties of architected materials by DOTS and GAD (baseline). (b) U-MAP 2D representation of input distribution from both methods.(c) Stimulated strain-stress curve of both methods. The inlet shows the density matrix. (d) Von Mises stress and hydrostatic pressure across the DOTS and GAD scaffolds. (e) Optimizing the electronic properties of CCAs by DOTS and MCMC (baseline). (f) U-MAP 2D representation of input distribution from both methods. (g) Fermi surface of DOTS-designed CCAs (bcc and fcc). (h) Fermi surface smearing. The smearings increase as the iteration increases. Inlets show the curves along a selected momentum path on the Fermi surface, a quantitative measure for describing the smearing. (i) Optimizing the PPIs using DOTS and other two methods. (j) Alphafold2 predicted complex, where the cyclic peptide is designed by DOTS. (k) Interaction diagram of DOTS peptide with the target protein. (l) NMSE loss history of DOTS, BO and TuRBO5 (n) Correlation index of between the phase reconstructions (DOTS, expert, BO and TuRBO5) and ground-truth. (m) The simulated and DOTS-reconstructed transmission functions, where the 1D distance profiles are shown.