Physics Informed Bayesian Machine Learning of Sparse and Imperfect Nuclear Data

The prevailing data-driven machine learning has been plagued by the absence of physics knowledge and the scarcity of data. We implement the physics-model informed prior into Bayesian machine learning to evaluate the energy dependence of independent fission product yields, which are crucial for advanced nuclear energy applications but only sparse and imperfect experimental data are available. The informative prior is the posterior after learning the generated data from fission models. Furthermore, cumulative fission yields are included as a physical constraint via a conversion matrix to provide augmented energy dependence. Our work demonstrated a truly Bayesian machine learning by incorporating comprehensive physics knowledges as a powerful tool to exploit the sparse but expensive nuclear data.

💡 Research Summary

The paper addresses a fundamental challenge in nuclear data science: how to reliably predict the energy‑dependent independent fission product yields when experimental measurements are extremely sparse, noisy, and often incomplete. Conventional data‑driven machine‑learning approaches suffer from two major drawbacks in this context: they lack embedded physical knowledge, leading to unphysical extrapolations, and they cannot compensate for the paucity of data, resulting in large uncertainties or over‑fitting.

To overcome these issues, the authors propose a physics‑informed Bayesian neural network (BNN) framework that integrates a physics‑based prior derived from the GEF (General Fission) model and enforces a physical constraint linking independent yields to cumulative yields via a conversion matrix. The workflow consists of three key stages:

1. Generation of a physics‑based prior – The GEF model, with its default parameter set, is used to generate a large synthetic dataset (≈9 300 points) of independent neutron‑induced fission yields for ^235U over a range of incident neutron energies (0, 0.6, 2, 4, 6, 8, 10, 12, 14 MeV). This synthetic data is used to train a BNN for 10⁵ steps, producing a posterior distribution of network weights P(w₁|D_phys). This posterior is then adopted as the prior P(w₂) for the subsequent inference on real experimental data.

2. Bayesian updating with experimental data – The experimental dataset comprises 587 independent‑yield points and 1 521 cumulative‑yield points extracted from the EXFOR library, together with evaluated values from JENDL‑5 at three benchmark energies (thermal, 0.5 MeV, 14 MeV). Using the physics‑informed prior, the BNN is updated via Bayes’ theorem, yielding a posterior weight distribution P(w₂|D_expt). Markov‑chain Monte‑Carlo (MCMC) sampling provides a full probability distribution for the network outputs, from which 95 % confidence intervals are directly obtained.

3. Physical constraint via conversion matrix – Independent and cumulative yields share the same fragment mass (A) and charge (Z) and are linked by a conversion matrix that accounts for β‑decay chains. The loss function therefore contains two terms: a χ² term measuring the discrepancy between predicted and measured independent yields, and a penalty term enforcing consistency between the network‑predicted independent yields and the cumulative yields after transformation. The global noise scale σ² is learned jointly with the network parameters, allowing the model to adaptively weight the different data sources.

The neural network architecture is modest (two hidden layers, 22 neurons each, tanh activation) but sufficient because the physics‑informed prior already encodes much of the complex nuclear physics (shell effects, odd‑even staggering, multi‑chance fission). The authors emphasize that the prior acts similarly to transfer learning: it dramatically accelerates convergence and reduces the risk of over‑fitting, as demonstrated in the loss‑versus‑epoch plots.

Results – When comparing three scenarios—(i) uninformed learning (standard BNN with only experimental data), (ii) informed learning (physics‑informed prior, no constraint), and (iii) informed learning with the cumulative‑yield constraint—the informed approaches consistently outperform the uninformed one. The normalization error drops from ~5.3 % (uninformed) to ~0.22 % (informed). Energy‑dependent trends are physically sensible: the symmetric fission channel grows exponentially with neutron energy, and odd‑even staggering in charge yields diminishes as excitation energy increases, both of which are captured only by the informed models.

Loss curves show that the physics‑informed prior yields rapid convergence within a few hundred training steps, whereas the uninformed model requires many more steps and exhibits larger fluctuations. Adding the cumulative‑yield constraint further reduces the loss by roughly 50 % and stabilizes the training, although residual discrepancies remain due to inconsistencies among the experimental cumulative‑yield data themselves.

Detailed examinations of specific isotopic chains (e.g., Z = 53 isotopes at 3 MeV and 8 MeV) reveal that the uninformed model underestimates yields and exhibits larger uncertainties, especially at higher energies. The informed model aligns well with the sparse experimental points at 3 MeV and provides smoother, more realistic energy dependencies. For isotopes ^135I and ^136I, the informed model predicts a non‑monotonic energy dependence for ^135I, reflecting the influence of the limited experimental data at low energy, while all models agree on a relatively flat trend for ^136I.

Conclusions and Outlook – The study demonstrates that embedding comprehensive physics knowledge into Bayesian machine learning—through a physics‑derived prior and explicit physical constraints—enables accurate, uncertainty‑quantified predictions of nuclear observables even when the underlying data are extremely scarce and noisy. The approach is readily extensible: more sophisticated fission models (e.g., TD‑GCM, Langevin dynamics) could be used to generate priors, and additional physical constraints (e.g., conservation of total mass, energy balance) could be incorporated. The methodology holds promise for a wide range of nuclear‑data applications, including fuel‑cycle simulations, medical isotope production planning, and the development of next‑generation reactor designs where reliable fission‑product yield data are essential.