Exploiting Gradients in Bayesian Inference of Expensive Simulators

Šimon Soldát; Václav Šmídl

cs.LGOct 8, 2026

Exploiting Gradients in Bayesian Inference of Expensive Simulators

Authors: Šimon Soldát, Václav Šmídl

Organizations: Department of Computer Science, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic · Institute of Plasma Physics of the Czech Academy of Sciences, Prague, Czech Republic

Abstract

Simulators based on differential equations are ubiquitous in science and engineering. They are often used in simulation-based inference to evaluate the posterior distribution of the input parameters based on real-world observations of the simulator outputs. However, inference becomes challenging when individual simulator evaluations are computationally expensive. In such cases, a Bayesian optimization-based active learning approach with Gaussian process surrogate models has been used to maximize the information obtained from a limited simulation budget. Recently, gradients of simulator outputs with respect to input parameters have become increasingly available, yet they are rarely exploited for inference. Even though we only need to learn the simulator input-output relationship, gradient information can provide an additional valuable signal to guide the active learning procedure. This is of particular interest in the case of expensive simulators, when sample efficiency is crucial. In this paper, we demonstrate how incorporating gradient information into the Gaussian process surrogate accelerates Bayesian optimization-based inference under a limited simulation budget. Our results show significant improvement in convergence speed from using gradient information. For reverse-mode differentiation, the inference efficiency gains are maintained when accounting for the additional computational cost. In contrast, for forward-mode differentiation, the inference speed-up does not outweigh the computational costs. These results indicate that gradient-enhanced surrogates are beneficial primarily in problems where the number of parameters exceeds the output dimensionality, where reverse-mode differentiation is efficient.

Figures & tables

Panel of Figure 1 — Fig. 1 : Comparison of BO (left) and BOSIP (right) on trivial example with uniform prior $p(x)$ , linear simulator $y=x$ , normal likelihood $p(z|y)=\mathcal{N}(y,0.2)$ , and observation $z_{o}=0$ . The MAP solution is $\hat{x}=0$ and the full Bayesian posterior is $p(x|z_{o})=N(0,0.2)$ . The classic BO-based approach focuses data at the mode of the posterior in order to find the maximizer $\hat{x}$ , while BOSIP spreads data across the high-probability regions in order to learn the shape of the posterior $p(x|z_{o})$ .

Figure 2

Fig. 2 : Comparison of BOSIP with and without gradients across seven benchmarks. The solid lines show median scores across 20 runs. The dashed lines show the scores adjusted for the additional computational cost of the gradients. The dotted lines show the score adjusted for the total information obtained from the simulator. Note that the cost-adjusted and data-adjusted scores are equal in case of forward-mode gradients. Importantly, when reverse-mode is used, the performance of the gradient-enabled BOSIP is better than the gradient-free BOSIP even when accounting for the additional computational cost.

Fig. 3 : Performance on higher-dimensional benchmarks. The performance benefits of exploiting reverse-mode gradients are larger in higher dimensions. For the same cost of a reverse-mode gradient pass, we obtain more gradient information, improving inference efficiency further.

Fig. 4 : Impact of suitable proxy variable design on the SIR benchmark. The response surface of the SIR simulator exhibits sharp ridges, where the gradients attain huge values, causing numerical issues in the gradient-enhanced surrogate. The performance is greatly improved by using a proxy variable that smooths the response surface, significantly mitigating the numerical issues. The gradient-enhanced runs are truncated to 50 simulations, as the numerical issues impede further convergence.

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