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Explaining Process Control Optimisation Recommendations via GradientSHAP and Implicit Differentiation

This paper proposes a real-time explainable AI framework that combines Implicit Function Theorem-based sensitivity analysis with SHAP attribution and Large Language Models to generate accurate, natural language explanations for industrial process control optimization, achieving over 40x speedup compared to traditional methods while maintaining high fidelity in operator-facing insights.

Original authors: Paul Darm, Cem Alpturk, Kenneth Ulrich, William Duncan, Ali Anwar, Annalisa Riccardi

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Paul Darm, Cem Alpturk, Kenneth Ulrich, William Duncan, Ali Anwar, Annalisa Riccardi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where giant machines run factories, but instead of human hands turning the dials, a super-smart computer brain makes all the decisions. This is the realm of industrial process control, where algorithms constantly tweak settings like pressure and speed to keep production running smoothly and safely. But here's the tricky part: the people who actually run the machines—the operators—often don't trust the computer. If a robot suddenly says, "Turn the pressure up!" the human operator might hesitate, thinking, "Why? Is it broken? Is it going to explode?" This is the trust gap.

To fix this, scientists have been using a tool called Explainable AI (XAI). Think of XAI as a translator that turns the computer's secret math into plain English. One popular translator is called SHAP, which acts like a detective, breaking down a decision to show exactly which clues (or input numbers) led to the final verdict. However, there's a catch: explaining the decisions of an optimization algorithm (a computer that constantly searches for the "perfect" setting) is incredibly slow and hard to do in real-time. It's like trying to explain a chess move by simulating every possible future game in your head before you can speak. This paper tackles that speed problem, aiming to give operators instant, clear reasons for the machine's choices so they can trust and act on them immediately.

The Paper's Big Idea: The "Instant Translator" for Machine Brains

This paper introduces a clever new way to explain why industrial machines change their settings, making the process fast enough to happen in real-time. The authors, a team of researchers and engineers from the University of Strathclyde and Weir, focused on a specific piece of heavy machinery called a High Pressure Grinding Roll (HPGR). Imagine two giant, counter-rotating rollers crushing rocks into powder. The machine needs to adjust two main knobs: how hard it squeezes (pressure) and how fast it spins (speed).

The computer running the show constantly looks at the rocks coming in (their size and shape) and the goals set by the humans (how much powder to make and how fine it should be). It then calculates the perfect pressure and speed to hit those targets. But when the computer changes its mind—say, it decides to lower the pressure by 3 bars—the operator needs to know why immediately.

The Old Way vs. The New Way

Previously, to explain these changes, the computer had to use a method called KernelSHAP. This was like asking the computer to run the entire optimization problem over and over again, changing one tiny thing at a time, to see how the result shifted. It was accurate but painfully slow. The paper shows that this old method took about 362.9 seconds (nearly 6 minutes) to generate a single explanation. In a real factory, waiting that long is useless; the machine needs an answer now.

The authors' new approach uses a mathematical shortcut called the Implicit Function Theorem (IFT). If you imagine the optimization problem as a complex maze, the old method tried to walk every path to find the exit. The new method uses a map (the math of the maze itself) to instantly calculate exactly how the exit moves if you shift a wall. This allows them to use a faster version of the translator called GradientSHAP.

The Results: Speed and Clarity

The team tested their new method on the HPGR machine. The results were dramatic:

Speed: The new method generated explanations in just 7.7 to 8.6 seconds. That is a 40 to 47 times speedup compared to the old method.
Accuracy: Despite being so much faster, the new method was almost perfectly accurate. When they compared the new "instant" explanations to the old "slow" ones, the results matched with a correlation of greater than 0.99. This means the fast method didn't cut corners on truth; it just found a smarter path.

From Math to Stories

Getting the numbers right is only half the battle. The paper also shows how to turn those numbers into a story that a human operator can understand. They used a Large Language Model (LLM)—a type of AI that is great at writing—to take the raw math data and write a natural language explanation.

However, the authors learned a valuable lesson from talking to real engineers: don't overwhelm the user with math.

What didn't work: Showing the raw SHAP numbers (like "0.15 bar contribution") confused the operators. One expert noted, "As someone with no background in SHAP, the section 'What These SHAP Values Mean' isn't clear to me."
What worked: The best explanations started with a one-sentence summary of the action. Then, they explained the reason in plain terms, using rounded numbers and percentages that felt familiar.

For example, in a test scenario where the machine needed to produce slightly coarser rocks with lower throughput, the AI generated this explanation:

"Because the operator wants lower throughput and a slightly coarser product, the system recommends backing off both pressure and roll speed, even though the feed itself is coarser."

It then broke it down simply: "The operator allows the product to be about 10 µm coarser, so the HPGR does not need to crush as hard... Net result: pressure decreases by 3 bar."

What This Means for the Future

The paper confirms that it is possible to have both speed and trust in automated factories. By using the math of the optimization problem itself (rather than treating it as a black box), the team proved that you can get real-time, human-friendly explanations for complex machine decisions.

There are still some limits, though. The current method works best when the machine's solution is "inside" the safe zone (unconstrained). If the machine hits a hard wall or a safety limit, the math gets trickier, and the authors note that future work is needed to handle those specific boundary cases. But for now, this approach offers a promising bridge between the cold logic of algorithms and the practical needs of the humans who keep the world's factories running.

Technical Summary: Explaining Process Control Optimisation via GradientSHAP and Implicit Differentiation

Problem Statement
While automated mathematical optimisation is increasingly adopted in industrial processes (e.g., mineral processing) to enhance efficiency and safety, a "trust gap" persists between algorithm designers and the operators who must execute recommendations. Operators often distrust automated outputs they cannot interpret. While Explainable AI (XAI) methods like SHAP (SHapley Additive exPlanations) have successfully improved the interpretability of machine learning predictions, applying similar techniques to optimisation outputs remains nascent. Existing approaches for explaining optimisation often rely on black-box sampling methods (e.g., LIME-inspired sampling or KernelSHAP) that require repeated, computationally expensive optimisation solves for every explanation, making real-time application infeasible.

Methodology
The authors propose a pipeline that integrates the Implicit Function Theorem (IFT) with GradientSHAP and Large Language Models (LLMs) to generate efficient, natural language explanations for optimisation recommendations.

Implicit Differentiation for Sensitivity Analysis:
The core innovation lies in treating the optimisation problem as a differentiable component. For an unconstrained optimisation problem $\min_p L(p, s)$ , where $p$ represents control parameters and $s$ represents inputs/targets, the optimal solution $p^*(s)$ is implicitly defined by the first-order optimality condition $\nabla_p L(p^*, s) = 0$ .
Using the Implicit Function Theorem, the authors derive the exact sensitivity of the optimal parameters with respect to the inputs:
$\frac{\partial p^*}{\partial s} = -H^{-1} \cdot \frac{\partial^2 L}{\partial p \partial s}$
where $H$ is the Hessian of the loss function. This allows for the computation of exact gradients $\partial p^*/\partial s$ via automatic differentiation (using JAX) with a single optimisation solve, rather than requiring thousands of re-solves.
GradientSHAP for Attribution:
Leveraging the sensitivities computed via IFT, the authors apply GradientSHAP to compute feature attributions. Unlike KernelSHAP, which approximates Shapley values through Monte Carlo sampling of the model (requiring many optimisation solves), GradientSHAP integrates gradients along a path from a baseline to the instance of interest. Because the gradients are available analytically via IFT, this method achieves high sample efficiency.
Narrative Generation:
The resulting SHAP attributions (quantifying the contribution of each input feature to the change in optimal parameters) are fed into an LLM (OpenAI GPT-5). The LLM is prompted with domain knowledge regarding High Pressure Grinding Rolls (HPGR) physics to translate numerical attributions into operator-friendly narratives. Crucially, the LLM acts as a translator of the SHAP values, not a reasoner about the physics itself, ensuring the explanation remains grounded in the actual optimiser's output.

Key Contributions

GradientSHAP for Optimisation: The paper presents the first application of GradientSHAP to explain optimisation algorithm outputs. By deriving gradients from the Implicit Function Theorem, the method bypasses the need for repeated optimisation solves.
Efficiency and Accuracy: The approach achieves equivalent SHAP attributions to KernelSHAP (correlation >0.99) with a computational speedup of over 40×. This efficiency enables real-time generation of explanations.
Human-Centric Validation: The study validates the approach on an industrial HPGR control problem (22 features) and incorporates feedback from domain experts to refine the explanation format. The authors found that effective explanations must lead with a one-sentence actionable summary, omit raw SHAP values, and use rounded, operator-friendly units (e.g., percentages) rather than raw technical data.

Results

Computational Performance: In a comparative study, KernelSHAP required approximately 100 samples to reduce variance, taking ~38 seconds. In contrast, GradientSHAP produced stable results with as few as 2–3 path integration samples, taking ~8 seconds. This resulted in a speedup of 45.3× to 47.1× relative to a converged KernelSHAP baseline (1000 samples).
Attribution Validity: Correlation analysis across 72 baseline-instance pairs showed a correlation coefficient $r > 0.99$ between GradientSHAP and converged KernelSHAP for scalar inputs. For array-valued inputs (20-dimensional Particle Size Distribution), the methods produced nearly identical attribution profiles, correctly identifying that mid-range particle sizes have the strongest influence on optimal pressure and speed settings.
Qualitative Feedback: Three domain experts (process engineers and an HPGR specialist) evaluated the generated narratives. They confirmed the explanations were accurate and helpful but emphasized the need for a "summary-first" structure and the removal of technical jargon (like raw SHAP values) to ensure practical utility.

Significance and Claims
The paper claims to bridge the gap between automated optimisation and operator trust by providing a scalable, mathematically rigorous method for explaining optimisation decisions. By exploiting the mathematical structure of optimisation (via IFT) rather than treating the optimiser as a black box, the authors demonstrate that high-fidelity explanations can be generated in real-time. The methodology is presented as broadly applicable to differentiable optimisation in process control, though the current implementation is limited to unconstrained problems where solutions lie in the interior of the feasible region. Future work is identified as extending the approach to constrained optimisation (handling active set changes via KKT systems) and conducting formal user studies.