Description
Equations of state (EoS) are essential to close the conservation equations in hydrodynamic simulations commonly used in many fields of application, such as inertial confinement fusion (ICF) [1] and planetary science [2]. They set a relationship between the state variables of a medium and are generally constructed on a given grid in density-temperature by a chosen first-principles-based model [2-4]. Such tabulated EoS are called in hydrodynamics simulations through a numerical interpolator to access energy and pressure at any given point in density -temperature. Traditional interpolators like rational polynomials [5], Hermite splines [6, 7] and b-splines [8] ensure thermodynamic consistency when used on the Helmoltz free-energy, but lack the flexibility of machine learning (ML) methods when it comes to integrating data from several sources.
In recent years, the usage of ML in plasma science has risen exponentialy [9, 10]. Artificial neural networks (ANN) have consequently been explored as a novel interpolation tool for EoS [11-15]. A promissing ANN architecture is one that leverages automatic differentiation (AD) to access derivatives of the ANN [13, 14] and therefore enforce thermodynamic consistency during training by interpolating the free-energy with knowledge on its partial derivatives. In such a case, the ANN predicts the Helmoltz free-energy infered from other available state variables, and is therefore inherently thermodynamically consistent but does not explicitely preserve thermodynamic stability.
In this ongoing work, we aim to develop a thermodynamic stability-constrained ANN framework that predicts the Helmholtz free-energy while enforcing thermodynamic inequalities (e.g., convexity of entropy) via the loss function. This builds on our prior work using automatic differentiation (AD) to guarantee consistency but now explicitly targets stability to prevent unphysical behaviors such as negative heat capacities. Our methodology involves training ANN to infer the free-enery from internal energy and pressure, in addition to augmenting the loss function with penalties for violations of thermodynamic stability criteria, and lastly, benchmarking against traditionnal interpolators, checking for accuracy within the tabulated grid, as well as comparing unphysical extrapolations in off-grid regions. Future work includes comparing our hard constrain architecture to Physics Informed Neural Networks (PINN), and integrating experimental constraints to further ground the model in measurable physics, as well as extending stability guarantees to multi-phase EoS tables.
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[15] G. Chaparro, and E. A. Müller 2023 The Journal of Chemical Physics 158.18