🤖 machine learning

Steering Neural Network Training through Interpretable Constraints Based on Partial Dependence

This paper introduces an explanation-guided learning approach that steers neural network training using partial dependence constraints to align model behavior with domain knowledge, demonstrating improved performance, data efficiency, and interpretability on regression tasks compared to unconstrained models.

Original authors: Yann Claes, Pierre Geurts, Vân Anh Huynh-Thu

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yann Claes, Pierre Geurts, Vân Anh Huynh-Thu

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 you are teaching a super-smart, but slightly mischievous, robot to predict the weather. You give it mountains of data, and it learns to guess the temperature. But here's the catch: the robot is a "black box." It gets the right answer, but sometimes for the wrong reasons. Maybe it's learned that "clouds mean rain" when actually, it's the humidity that matters, and it just happened to see clouds and rain together a lot.

This is the problem the authors, Yann Claes and his team, are tackling. They want to stop the robot from guessing and start it from understanding based on what humans already know.

The "Partial Dependence" Compass

The authors introduce a new way to steer these neural networks (the robots) using something called Partial Dependence (PD). Think of PD as a "marginal effect" map. If you ask, "How does the temperature change only because of the humidity, ignoring everything else?" the PD map shows you that specific relationship.

Usually, we just look at these maps after the robot learns to see if it made sense. But this paper suggests a different approach: Explanation-Guided Learning. Instead of waiting to see if the robot is right, you hand it a compass before it starts learning. You say, "Hey robot, we know that humidity and temperature have a specific relationship (like a curve or a straight line). Make sure your internal map looks like this."

The "Linspace" vs. "In-Distribution" Showdown

The team tested two ways of giving this compass to the robot:

In-Distribution Constraints: This is like telling the robot, "Make sure your map looks right only in the places where we have data." If you only have weather data from summer, the robot only promises to get the summer relationship right.
Linspace Constraints: This is the paper's star player. Here, you tell the robot, "Make sure your map looks right across a whole grid of evenly spaced points, even in places where we have no data." It's like drawing a smooth, perfect curve across the entire map, not just the spots where you dropped a pin.

The Big Finding:
In their experiments—which included predicting concrete strength, water chemistry, and even the swing of a damped pendulum—the Linspace method was the clear winner.

When the robot was trained with very little data (sometimes as few as 25 or 50 samples), the "In-Distribution" robot got confused and forgot the rules outside the data it saw. But the Linspace robot? It kept the shape of the relationship correct, even when it had to guess about weather conditions it had never seen before.

For example, in the concrete strength experiment, they knew that as concrete gets older, its strength grows in a specific logarithmic way.

The Unconstrained robot (no compass) and the In-Distribution robot (limited compass) both failed when asked to predict strength for very old concrete (data outside the training range). They forgot the curve.
The Linspace robot, however, remembered the shape. It predicted the strength of old concrete much better, even though it had only seen young concrete during training.

What They Explicitly Rule Out

The authors are careful to point out what their method is not.

It's not "Physics-Informed" in the traditional sense: They compared their method to other techniques that try to mix physics equations directly into the code (like adding a physics formula to the robot's brain). They found that if the real world is messy and not perfectly additive (meaning the parts don't just add up simply), those physics-mixing methods can fail. Their method is more flexible because it doesn't force the whole model to be a simple sum of parts; it just forces the shape of the relationship to be right.
It's not magic: If you give the robot a compass with the wrong direction (mis-specified knowledge), it won't work perfectly. However, they found that even a vague or slightly wrong compass (like guessing a square root relationship instead of a logarithmic one) was still better than having no compass at all when the data was scarce.

How Sure Are They?

The authors are very confident in their results, but they are specific about where that confidence comes from.

They measured this on real-world datasets (like concrete strength and water pH) and simulated it on synthetic problems (like the famous "Friedman problem" and a swinging pendulum).
They didn't just say "it looks good." They ran the numbers over and over (10 different training starts for synthetic problems, 20 splits for real data) and showed that the Linspace method consistently had lower errors and better "maps" (PD estimates) than the others.
They explicitly state that while the method takes a bit more computing time (because the robot has to check its map against the grid constantly), it is worth it for the accuracy, especially when you don't have a lot of data.

The Takeaway for a Curious Teen

Imagine you are learning to draw a mountain.

The Unconstrained Robot looks at a few photos of mountains and tries to guess the shape. If it hasn't seen a snow-capped peak, it might draw a flat hill.
The In-Distribution Robot promises to draw the mountains exactly as they appear in the photos it has. But if you ask it to draw a mountain it hasn't seen, it freezes or draws something weird.
The Linspace Robot is given a rule: "Mountains always go up and then down in a smooth curve." Even if it only sees the bottom of the mountain in the photos, it uses that rule to draw the peak correctly.

The paper suggests that by forcing our AI to follow these "smooth curve" rules (Partial Dependence constraints) across the whole map, not just the parts we've seen, we can build smarter, more reliable models that don't get lost when the data gets scarce or the situation changes. It's a way of teaching the robot to respect the "laws of the land" before it even starts exploring.

Technical Summary: Steering Neural Network Training through Interpretable Constraints Based on Partial Dependence

Problem Statement

The field of Explainable Artificial Intelligence (XAI) has seen significant growth in developing techniques to interpret model decisions. However, a critical gap remains in assessing the quality of these explanations and, more importantly, in adjusting models to ensure their explanations align with prior domain knowledge. Existing approaches in "explanation-guided learning" (EGL) primarily focus on classification tasks (often involving images or text) and typically assume knowledge regarding feature importance or specific input regions. Furthermore, many methods that incorporate constraints focus on structural properties of the function (e.g., convexity, monotonicity) rather than the explanations themselves.

This work addresses the challenge of steering neural network training for regression problems—specifically including dynamical systems forecasting—to ensure that the model's average response to certain features (its partial dependence) aligns with specific functional domain knowledge. The goal is to produce models that are not only more data-efficient and robust to out-of-distribution (OOD) shifts but also yield explanations faithful to the provided prior knowledge.

Methodology

Partial Dependence as a Constraint

The authors define the problem as learning a differentiable model $h_\theta(x)$ such that its partial dependence (PD) on a subset of features $x_k$ aligns with a known functional form. The PD of a function $f$ on $x_k$ is defined as the expected value of the function over the marginal distribution of the remaining features $x_{-k}$ :
$PD_\infty(f, x_k) = \mathbb{E}_{x_{-k}}[f(x_k, x_{-k})]$

The authors assume that the true PD follows an algebraic shape $h^\phi_k(x_k)$ , parameterized by $\phi$ . This shape may represent a known physical relationship (e.g., logarithmic, sinusoidal) up to scaling and bias constants.

Training Algorithm

The training objective minimizes a combined loss function $L_\lambda$ that balances predictive accuracy and adherence to the PD constraint:
$L_\lambda(h_\theta, h^\phi_k, f; LS) = (1 - \lambda)L_{tot}(h_\theta, f; LS) + \lambda L_{PD}(h_\theta, h^\phi_k; LS)$
Where:

$L_{tot}$ is the standard mean squared error on the training set.
$L_{PD}$ measures the squared difference between the estimated PD of the model and the target function $h^\phi_k$ evaluated on a specific set of points $x_k$ .
$\lambda \in [0, 1]$ controls the trade-off.

Optimization Strategy:
The authors propose an alternating optimization procedure (Algorithm 1):

Initialization: The target function $h^\phi_k$ is fitted to the raw outputs for a set number of epochs to initialize $\phi$ .
Alternating Steps: The model parameters $\theta$ and the prior parameters $\phi$ are updated in alternating blocks. Every $S$ epochs, $\phi$ is updated to minimize the PD loss given the current $\theta$ , and then $\theta$ is updated to minimize the combined loss.
Dataset Construction: A crucial component is the construction of a separate dataset $LS_{PD}$ $L S_{P D}$ for the constraint term. This dataset can be either:
- In-distribution (ID): Samples drawn from the training distribution $p(x_k)$ .
- Linspace (L): A set of evenly spaced values covering the input domain (potentially extending beyond the training distribution). The authors argue that constraining on evenly spaced points helps the model generalize to poorly sampled regions and OOD data.

Key Contributions

Novel EGL Framework for Regression: Unlike previous EGL works focused on feature importance in classification, this approach constrains the functional form of the partial dependence in regression tasks, including dynamical systems.
Linspace Constraint Configuration: The paper introduces and empirically validates the use of "linspace" constraints (constraining PD on evenly spaced points across the domain) as a superior alternative to in-distribution constraints. This configuration forces the model to learn the correct functional shape even in regions with sparse training data.
Robustness to Data Scarcity and OOD Shifts: The method is shown to be particularly effective in low-data regimes and when test-time data shifts outside the training distribution.
Comparison with Physics-Informed ML: The authors compare their single-model constraint approach against physics-informed hybrid models (e.g., APHYNITY, PD-hybrid). They demonstrate that their method often yields better predictive performance and more faithful PD estimates, particularly when the underlying problem does not strictly satisfy the additivity assumptions required by hybrid models.

Experimental Results

The authors evaluated the method on synthetic problems (Friedman, Product), real-world datasets (Concrete Compressive Strength, PHALK), and a dynamical system (Damped Pendulum).

Predictive Performance: Models trained with PD constraints consistently outperformed unconstrained models, especially as the training sample size decreased (e.g., 50 samples). In many cases, constrained models trained on 50 samples achieved performance comparable to unconstrained models trained on 300 samples.
Explanation Quality: The PD estimates of constrained models aligned significantly better with the prior knowledge (ground truth or approximated) than those of unconstrained models.
Linspace vs. In-Distribution: Across all experiments, linspace constraints (constraining on evenly spaced points) generally outperformed in-distribution constraints. This was particularly evident in extrapolation scenarios where test data fell outside the training domain. Linspace constraints helped dampen the impact of domain shifts, whereas in-distribution constraints often failed to generalize beyond the training range.
Robustness to Mis-specified Knowledge: Even when the prior knowledge was imperfect (e.g., assuming a square root relationship instead of logarithmic, or using a Fourier series with incorrect frequencies), the linspace-constrained models remained robust. They often outperformed unconstrained models and, in extrapolation settings, produced PD shapes closer to the true underlying function than models trained with in-distribution constraints.
Dynamical Systems: The method was successfully extended to dynamical systems forecasting (Neural ODEs), showing improved trajectory prediction and PD alignment, particularly when initial states were out-of-distribution.

Significance and Claims

The paper claims that steering neural networks via partial dependence constraints is a powerful mechanism for explanation-guided learning in regression. The primary significance lies in the ability to:

Improve Data Efficiency: Achieve high predictive accuracy with significantly fewer training samples by leveraging functional prior knowledge.
Enhance Generalization: Produce models that generalize better to out-of-distribution data by enforcing the correct functional shape across the entire input domain, not just where data is dense.
Ensure Faithful Explanations: Guarantee that the model's interpretability (via PD plots) reflects the domain expert's knowledge, preventing the model from learning "correct" predictions for "wrong" reasons.

The authors note that while the method incurs higher computational costs due to the estimation of partial dependence (requiring model evaluations over multiple feature combinations), this is a manageable trade-off for low-data regimes where generalization and interpretability are paramount. They conclude that this approach offers a more generic alternative to physics-informed hybrid models, as it does not require explicit additive decomposition of the problem.