BONSAI: Structure-exploiting robust Bayesian optimization for networked black-box systems under uncertainty

TL;DR

BONSAI tackles robust design under uncertainty for expensive simulators by modeling the objective as a function network of known and unknown components, and applying a two-stage Thompson sampling-based Bayesian optimization strategy. By placing Gaussian process priors on individual node functions and propagating uncertainty through the network, BONSAI achieves improved sample efficiency and robust performance compared to black-box baselines. A finite-time regret bound is established for the nominal (non-robust) setting, linking BONSAI to classical BO theory, while empirical results across synthetic and real-world process-system benchmarks demonstrate consistent gains, especially in modular or cyclic networks. The framework is practically impactful for uncertainty-aware design in complex engineering systems, offering a principled path to leverage partial structure without requiring full equation-based models.

Abstract

Optimal design under uncertainty remains a fundamental challenge in advancing reliable, next-generation process systems. Robust optimization (RO) offers a principled approach by safeguarding against worst-case scenarios across a range of uncertain parameters. However, traditional RO methods typically require known problem structure, which limits their applicability to high-fidelity simulation environments. To overcome these limitations, recent work has explored robust Bayesian optimization (RBO) as a flexible alternative that can accommodate expensive, black-box objectives. Existing RBO methods, however, generally ignore available structural information and struggle to scale to high-dimensional settings. In this work, we introduce BONSAI (Bayesian Optimization of Network Systems under uncertAInty), a new RBO framework that leverages partial structural knowledge commonly available in simulation-based models. Instead of treating the objective as a monolithic black box, BONSAI represents it as a directed graph of interconnected white- and black-box components, allowing the algorithm to utilize intermediate information within the optimization process. We further propose a scalable Thompson sampling-based acquisition function tailored to the structured RO setting, which can be efficiently optimized using gradient-based methods. We evaluate BONSAI across a diverse set of synthetic and real-world case studies, including applications in process systems engineering. Compared to existing simulation-based RO algorithms, BONSAI consistently delivers more sample-efficient and higher-quality robust solutions, highlighting its practical advantages for uncertainty-aware design in complex engineering systems.

Figures (12)

• Figure 1: Example function network for robust process systems design. The overall objective function $g(x, w)$ combines outputs from five interconnected modules: CFD simulations, process simulation software, life cycle analysis (LCA), experimentation-based product development, and economic post-processing. Design variables $\boldsymbol{x}$ (e.g., plant design choices, pathway selection) and uncertain inputs $\boldsymbol{w}$ (e.g., flow variability) propagate through a network of intermediate variables ($h_1$ through $h_5$), capturing modular structure and interactions among subsystems. BONSAI exploits this structure for robust, sample-efficient optimization in the presence of uncertainty.
• Figure 2: Comparison of white-box, black-box, and the function network (grey-box or hybrid) modeling approaches for the objective function $g(x, w) = -\left(f_1(x, w)^2 + f_2(x, w)^2\right)^2$ where $f_1(x, w) = x^2 + w - 5$ and $f_2(x, w) = x + w^2 - 1$, with $x \in [-2.5,2.5]$ and $w \in [-1,1]$. Top: Illustration of function representations used in each approach, reflecting different amounts of prior knowledge. Middle: Contour plots of the true objective (left) and the posterior mean surfaces from the black-box (center) and grey-box (right) models. Black dots indicate sample locations. Bottom: The projected objective function $G(x) = \min_{w} g(x, w)$, showing the true value (black), posterior mean (violet), and 95% confidence band (violet cloud). The grey-box model provides substantially improved prediction and uncertainty estimates compared to the black-box baseline. The red vertical line in the white-box plot indicates the true max-min solution.
• Figure 3: Function network structures for the synthetic test problems. Nodes correspond to component functions and directed edges indicate dependency. White-box aggregators are shown in white.
• Figure 4: Contour plots of the robust objective $G(\boldsymbol{x}) = \min_{\boldsymbol{w} \in \mathcal{W}} g(\boldsymbol{x}, \boldsymbol{w})$ for the Polynomial (left) and Modified Sine (right) problems. The robust optimum $\boldsymbol{x}^\star$ is marked by a star. These plots reveal the strong nonstationarity and multimodality that make these functions particularly difficult for existing black-box robust optimization methods.
• Figure 5: Function network representation of the heat exchanger network flexibility problem (left) and the robust robot pushing problem (right).

Theorems & Definitions (15)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Lemma 1
• Lemma 2: Posterior-sampling identity
• proof
• Lemma 3: Variance recursion and network bound
• proof
• Lemma 4: Node variance sum $\leq$ MIG