A Gaussian Process framework for constraining the nuclear equation of state from microscopic calculations with correlated uncertainties

We present constraints on the nuclear equation of state (EOS) from microscopic asymmetric matter calculations at zero temperature based on chiral nucleon-nucleon and three-nucleon interactions. The constraints include the saturation point, the isospin dependence of the incompressibility, and the symmetry energy, as well as the crust-core transition density of neutron-star matter. To quantify and propagate correlated uncertainties from noisy many-body calculations to derived observables, we introduce GPDiff, an efficient JAX-based Python package for multivariate Gaussian process (GP) regression with automatic differentiation. After training, GPDiff enables joint predictions of the EOS and derivatives of arbitrary order with respect to the input variables, including mixed partial derivatives. In this initial application, we analyze recent high-order many-body perturbation theory calculations of asymmetric matter up to about twice saturation density and explore nonstationary change-surface kernels, a class of input-dependent kernels, for modeling the EOS. GPDiff is broadly applicable to microscopic nuclear EOS calculations at zero and finite temperature and provides a versatile package for GP-based uncertainty quantification and inference of the nuclear EOS.

Improvements to AI Systems:

1. Uncertainty-Aware Multi-Output Regression with Derivative Constraints:

• Integrate GPDiff’s multivariate Gaussian process (GP) framework into AI systems that predict physical observables (e.g., nuclear EOS) from noisy, sparse data.

• Enable the AI to jointly predict the target function and its derivatives (including mixed partials) with correlated uncertainties, not just point estimates.

• The improved system can enforce physical consistency (e.g., thermodynamic relations) by training on derivative information, leading to more robust extrapolation beyond training data.

2. Nonstationary Kernel Learning for Phase-Transition Detection:

• Adopt change-surface kernels (input-dependent kernels) to model systems with abrupt regime changes (e.g., crust-core transition in neutron stars).

• The AI can automatically learn where the correlation structure changes, enabling it to detect phase boundaries or critical densities without explicit labels.

• This improves AI’s ability to segment heterogeneous data and make predictions with locally adaptive smoothness.

3. Automatic Differentiation for Physics-Informed Inference:

• Use GPDiff’s JAX-based automatic differentiation to embed GP predictions directly into larger differentiable pipelines (e.g., solving differential equations or optimizing experimental designs).

• The improved AI can backpropagate through the GP’s predictive mean and variance, enabling end-to-end training of downstream models (e.g., neutron-star mass-radius relations) while propagating uncertainty naturally.

4. Efficient Uncertainty Propagation in Many-Body Simulations:

• Replace costly Monte Carlo or bootstrap methods with GPDiff’s closed-form GP posterior to propagate correlated noise from microscopic calculations to macroscopic observables.

• The AI system can now produce calibrated error bars on derived quantities (e.g., symmetry energy, incompressibility) in real time, enabling rapid sensitivity analysis and model comparison.

5. Transfer Learning Across Temperatures and Densities:

• Leverage GPDiff’s flexibility to train on zero-temperature data and then fine-tune or condition on finite-temperature calculations, using the GP’s covariance structure to bridge regimes.

• The improved AI can predict EOS at arbitrary temperatures/densities with quantified confidence, even where data are sparse, by sharing information across correlated input regions.

What the Improved AI System Can Do:

• Given noisy, sparse microscopic nuclear data, it can output a full probabilistic EOS surface (pressure, energy density, etc.) with analytic derivatives and credible intervals, automatically detecting phase transitions.

• It can be embedded in astrophysical simulations to generate neutron-star mass-radius curves with rigorous uncertainty bands, directly informing gravitational-wave or X-ray observations.

• It can perform active learning: propose the next most informative density/temperature point to compute, minimizing predictive uncertainty under physical constraints.

• It can serve as a drop-in surrogate model for expensive many-body perturbation theory, accelerating nuclear structure calculations by orders of magnitude while preserving accuracy and error estimates.