Actively Learning Joint Contours of Multiple Computer Experiments

Technical Summary: Actively Learning Joint Contours of Multiple Computer Experiments

Problem Definition
The paper addresses the challenge of identifying an "optimal design point" $\tilde{x}$ within a design space $X \subset \mathbb{R}^d$ that simultaneously satisfies pre-specified target response values $\tau_r$ for multiple deterministic computer experiments $f^{(r)}$ (where $r = 1, \dots, R$). Unlike standard contour location (CL) which targets a single level set $f(x) = \tau$, this work targets the intersection of multiple contours: $\{ \tilde{x} \in X \mid f^{(r)}(\tilde{x}) = \tau_r, \forall r \}$. The motivation stems from high-fidelity computational fluid dynamics (CFD) simulations of a High-Speed Army Reference Vehicle (HARV), where the goal is to find fin configurations that result in zero pitch and roll moments (stable "trim" conditions) under specific flight parameters. The problem is characterized by expensive evaluations and limited computational budgets, necessitating sample-efficient active learning (AL) strategies.

Methodology: Joint Contour Location (jCL)
The authors propose a novel sequential design scheme termed "joint contour location" (jCL). Rather than merging exploration and exploitation into a single acquisition function, jCL employs two distinct mechanisms governed by a decision rule:

1. Exploitation Mechanism:

   • Objective: Aggressively target the intersection of contours.
   • Acquisition Function: Maximizes the joint posterior probability that all responses fall within a tolerance $t_n$ of their targets: $J_n(x, t_n) = \prod_{r=1}^R P(\tau_r - t_n \leq f^{(r)}_n(x) \leq \tau_r + t_n)$.
   • Adaptive Tolerance: A critical innovation is the adaptive shrinking of the tolerance $t_n$. It is defined as $t_n = w \cdot \min_{i} (\max_r |y_{ir} - \tau_r|)$, where $w < 1$. This ensures the tolerance tightens as better points are found, guides acquisitions toward the true solution, and prevents the algorithm from re-evaluating already observed points (which would yield a probability of 1 in a deterministic setting).

2. Exploration Mechanism:

   • Trigger: Activated when the maximum joint probability $\max_x J_n(x, t_n)$ falls below a user-defined threshold $p^*$ (e.g., 0.2), indicating low confidence in the current surrogate's ability to locate the solution.
   • Acquisition Strategy: Targets the Pareto front of the posterior standard deviations $\{\sigma^{(1)}_n(x), \dots, \sigma^{(R)}_n(x)\}$ across all functions.
   • Candidate Generation: To ensure candidates lie in high-uncertainty regions, the method proposes "triangulation candidates" (tricands) based on a Delaunay triangulation of existing data points. A candidate is selected from the Pareto front of these tricands to maximize uncertainty reduction across all responses simultaneously.

3. Surrogate Agnosticism:
   The procedure is model-agnostic, provided the surrogate can supply posterior predictive distributions. The authors demonstrate its application using independent Gaussian Processes (GPs), Multitask GPs (incorporating correlations between responses), and Deep Gaussian Processes (DGPs) for non-stationary surfaces.

4. Stopping Criteria:
   The design halts if the budget is exhausted or if a solution is found within a specified tolerance $\epsilon$. Crucially, the authors propose a natural stopping rule for cases where no solution exists: if a consecutive number of exploration steps are triggered without any progress toward exploitation, the design can be reasonably halted.

Key Contributions

• Novel Acquisition Framework: The separation of exploration and exploitation into distinct mechanisms with a decision rule, contrasting with previous works that attempt to balance both in a single function.
• Adaptive Tolerance: The introduction of a shrinking tolerance $t_n$ that dynamically adjusts based on observed data, preventing the "flatness" or "peakiness" issues common in fixed-tolerance probability-based acquisition functions.
• Joint Contour Strategy: A specific approach for targeting the intersection of multiple contours using Pareto-front-based exploration, addressing a gap in literature where methods for joint contours are scarce compared to single-response CL.
• Handling Non-Existence: A built-in mechanism to detect and halt searches when no solution exists, based on the frequency of triggered exploration steps.

Results
The authors validate jCL against six competing strategies: Latin Hypercube Sampling (LHS), alternating single-response CL methods (alternating entropy and alternating Pareto), Bayesian Optimization (BO) on a summed squared error function, zero-finding optimization, and a joint PDF-based method inspired by Graziani and Ngom (2024).

• Synthetic Benchmarks: Across 2D and 3D synthetic functions (Multimodal, Camelback, Gramacy, Ishigami), jCL consistently outperformed all competitors in terms of the squared Euclidean distance ($D_n$) between the best observed point and the true target. jCL achieved superior median performance with significantly fewer evaluations (e.g., reaching in 5–10 acquisitions what others achieved in 20+).
• Surrogate Performance: The method proved robust across different surrogate types (GP, DGP, Multitask GP). In the HARV application, both independent and multitask GPs successfully located the optimal fin configuration within 15 evaluations, whereas space-filling designs made little progress.
• Real-World Application: In the HARV CFD simulation, jCL efficiently identified fin configurations for stable flight (zero pitch/roll) with a budget of 25 evaluations, demonstrating its utility for building trimmed aerodynamic databases.

Significance and Claims
The paper claims that jCL provides a significant improvement over existing single-response CL strategies and optimization-based alternatives for identifying joint contours. The authors emphasize that their method is particularly effective because it avoids the inefficiency of learning entire contours when only their intersection is needed. They note that while the method is currently demonstrated in low-dimensional settings ($d \leq 3$) with small budgets, the framework is applicable to any surrogate providing posterior distributions. The authors modestly acknowledge limitations, such as the current focus on finding a single solution (rather than mapping multiple equivalent solutions) and the potential computational burden of triangulation candidates in high dimensions, suggesting future work in these areas. The procedure is presented as a plug-and-play solution for sequential design in multi-experiment settings.