Organizations: Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411008, India · Innovation and Research Division, Ge-Room Inc., 93160 Noisy le Grand, France · Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, 34151, Trieste, Italy∗ · Univ. Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France†
Interpreting machine-learning models has attracted increasing attention, particularly in the physical sciences, where one often seeks to understand the underlying mechanisms rather than merely make predictions. Multiple linear regression is often regarded as an interpretable alternative to more complex models, such as deep neural networks, because its predictions are expressed as explicit weighted sums of input features. However, when input features are strongly correlated, namely in the presence of multicollinearity, the learned weights can exhibit large dataset-to-dataset fluctuations and oscillatory behavior across physically similar features, making their interpretation difficult or even impossible. Although the instability of the weights under multicollinearity is well known in statistics, its consequences for physical interpretation, in particular its connection to oscillatory weights across physically similar features, have not been systematically clarified. Here, we theoretically discuss the mechanism behind this loss of interpretability by analyzing the eigenmodes of the feature correlation matrix. We show that small-eigenvalue modes associated with multicollinearity amplify fluctuations in the weights and generate oscillatory patterns that do not necessarily reflect meaningful contributions. We test this theoretical picture numerically on physics datasets and show that Ridge regularization suppresses these unstable modes, although the resulting weights must still be interpreted with caution. We further confirm the generality of our findings beyond physics by analyzing a diverse collection of publicly available datasets. Our results clarify why, in the presence of multicollinearity, physical interpretation can remain difficult even for linear regression models.
While there are many applications of ML to scientific problems that look promising, visuals can be deceiving. Using numerical analysis techniques, we rigorously quantify the accuracy, convergence rates, and generalization bounds of certain ML models applied to linear differential equations for parameter discovery or solution finding. Beyond the quantity and discretization of data, we identify that the function space of the data is critical to the generalization of the model. A similar lack of generalization is empirically demonstrated for commonly used models, including physics-specific techniques. Counterintuitively, we find that different classes of models can exhibit opposing generalization behaviors. Based on our theoretical analysis, we also introduce a new mechanistic interpretability lens on scientific models whereby Green's function representations can be extracted from the weights of black-box models. Our results inform a new cross-validation technique for measuring generalization in physical systems, which can serve as a benchmark.
Alejandro Francisco Queiruga, Theo Gutman-Solo, Shuai Jiang
We review the concepts of interpretability and explainability as they apply to machine learning in physics. We define interpretability as concerning the structural transparency of a model (the ability to understand or approximate its inner workings) and explainability as concerning the scientific content of a model (the ability to map it onto domain knowledge). We discuss the trade-offs each entails (interpretability vs. expressivity; explainability vs. adaptability), the contexts in which each is needed, and the intrinsic and post-hoc tools available for achieving them. Throughout, we emphasize that machine-learned models are subject to the same scientific questions as classical models, differing only in scale, and that interpretability and explainability are best understood as deliberate modeling choices rather than inherent properties. We also emphasize the importance of task specification and intervention plans as a core aspect of model design.
Understanding nonlinear feature interactions is crucial in science and engineering, yet standard multilayer perceptrons (MLPs) often capture such interactions only implicitly, leading to entangled representations that can impair robustness and interpretability. We investigate product-unit residual networks (PURe) that integrate multiplicative product units with residual connections to explicitly model cross-feature couplings while stabilizing optimization. We conduct a systematic evaluation on an interaction-driven synthetic benchmark and two real-world datasets, assessing predictive accuracy, robustness to Gaussian feature noise, and performance under limited training data, and we compare real- and complex-valued variants under a matched parameter budget. Beyond accuracy, SHapley Additive exPlanations (SHAP)-based interaction analyses show that PURe learns more concentrated and structurally coherent interaction patterns than MLP baselines. Overall, PURe achieves competitive or improved performance, better robustness and sample efficiency in low-data regimes, and enhanced interaction-level interpretability.