Physics-Constrained Neural ODEs for MXene Bandgap Prediction with Conformal Uncertainty

Two-dimensional transition metal carbides and nitrides, known collectively as MXenes, are attractive photocatalyst candidates because their surface chemistry and atomic composition can be tuned over a wide compositional window. A crucial design quantity is the electronic bandgap, which selects whether a given MXene couples with solar radiation and aligns with the redox levels of water splitting. High-fidelity bandgap calculations using the PBE0 hybrid functional are computationally expensive, which has motivated several machine learning surrogates. To the best of our knowledge, this is the first study to integrate a continuous-depth Neural Ordinary Differential Equation backbone with multi-fidelity Δ learning, distribution-free split-conformal calibration, and uncertainty-aware Pareto screening into a single mathematically grounded pipeline for MXene bandgap prediction. In this work, we develop a physics-constrained neural ordinary differential equation (PC-NODE) that predicts MXene bandgaps from a compact 34-dimensional descriptor set, without relying on the density of states. The model couples a classifier head for the metal/semiconductor decision with a regression head for the gap magnitude, and enforces three physically motivated properties: non-negativity of the predicted gap and monotonicity between the low-fidelity Perdew-Burke-Ernzerhof (PBE) and the high-fidelity PBE0 estimates are obtained exactly through a softplus-parameterised Δ learning construction, while a hurdle coupling that drives metal predictions towards zero is enforced via a quadratic penalty and verified empirically. In short, two of the three physical constraints are guaranteed by construction, and the third is approximately enforced and verified empirically; the same distinction is maintained consistently in the methodology, the constraint audit and the conclusion. Trained on the 4356-structure MXgap database, a ten-seed ensemble reaches a mean absolute error of 0.186 eV (per-seed 0.206±0.006 eV) and a coefficient of determination R2=0.880 on the semiconductor test subset, with a classifier accuracy of 0.856 and a Receiver Operating Characteristic Area Under the Curve (ROC-AUC) of 0.925. A split-conformal calibration step then delivers prediction intervals whose empirical coverage matches the 90% target within 0.5 percentage points. Finally, an uncertainty-aware Pareto screening step applies the trained surrogate to a held-out subset of 396 lanthanum-based MXenes and identifies 74 candidates inside the photocatalytic water splitting window [1.23, 3.10] eV. The framework offers a mathematically grounded, data-efficient alternative to feature-heavy pipelines and is reproducible from the open MXgap resource.