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Physics-informed neural networks (PINNs) are an emerging technology that can be used both in place of and in conjunction with conventional simulation methods. In this paper, we used PINNs to perform a forward simulation without leveraging known data. Our simulation was of a 2D natural convection-driven cavity using the vorticity-stream function formulation of the Navier-Stokes equations. We used both 2D simulations across the x and z domains at constant Rayleigh ( Ra ) numbers and 3D simulations across the x , z and Ra domains. The 3D simulation was tested for a PINN’s ability to learn solutions in a higher-dimensional space than standard simulations. The results were validated against published solutions at Ra values of 10 3 , 10 4 , 10 5 , and 10 6 . Both the 2D simulations and 3D simulations successfully matched the expected results. For the 2D cases, more training iterations were needed for the model to converge at higher Ra values (10 5 and 10 6 ) than at lower Ra (10 3 and 10 4 ) indicating increased nonlinear fluid-thermal coupling. The 3D case was also able to converge but, but it required more training than any of the 2D cases due to the curse of dimensionality. These results showed the validity of standard simulations via PINNs and the feasibility of higher-order parameter space solutions that are not possible using conventional methods. They also showcased the additional computational demand associated with increasing the dimensionality of the learned parameter space.

A physics-constrained deep learning surrogate that predicts the exponential ‘avalanche’ growth rate of runaway electrons (REs) for a plasma containing partially ionized impurities is developed. Specifically, a physics-informed neural network (PINN) that learns the adjoint of the relativistic Fokker–Planck equation in steady-state is derived, enabling a rapid surrogate of the RE avalanche for a broad range of plasma parameters, motivating a path towards an machine learning-accelerated integrated description of a tokamak disruption. A steady-state power balance equation together with atomic physics data is embedded directly into the PINN, thus limiting the PINN to train across physically consistent temperatures and charge state distributions. This restricted training domain enables accurate predictions of the PINN while drastically reducing the computational cost of training the model. In addition, a novel closure for the relativistic electron population used when evaluating the secondary source of REs is developed that enables improved accuracy compared to a Rosenbluth–Putvinski source. The avalanche surrogate is verified against Monte Carlo simulations, where it is shown to accurately predict the RE avalanche growth rate across a broad range of plasma parameters encompassing distinct tokamak disruption scenarios.

In a post-thermal-quench plasma, mitigated or unmitigated, the plasma power balance is mostly between collisional or Ohmic heating and plasma radiative cooling. In a plasma of atomic mixture n α with α labeling the atomic species, the power balance sets the plasma temperature, ion charge state distribution n α i with i the charge number, and through the electron temperature T e and ion charge state distribution n α i , the parallel electric field E ∥ . Since the threshold electric field for runaway avalanche growth E a v is also set by the atomic mixture, ion charge state distribution and its derived quantity, the electron density n e , the plasma power balance between Ohmic heating and radiative cooling imposes a stringent constraint on the plasma regime for avoiding and minimizing runaways when a fusion-grade tokamak plasma is rapidly terminated.

A physics-informed neural network (PINN) is used to evaluate the fast ion distribution in the hot spot of an inertial confinement fusion target. The use of tailored input and output layers to the neural network is shown to enable a PINN to learn the parametric solution to the Vlasov-Fokker-Planck equation in the absence of any synthetic or experimental data. As an explicit demonstration of the approach, the specific problem of Knudsen layer fusion yield reduction is treated. Here, predictions from the Vlasov-Fokker-Planck PINN are used to provide a non-perturbative solution of the fast ion tail in the vicinity of the hot spot thus allowing the spatial profile of the fusion reactivity to be evaluated for a range of collisionalities and hot spot conditions. Excellent agreement is found between the predictions of the Vlasov-Fokker-Planck PINN and results from traditional numerical solvers with respect to both the energy and spatial distribution of fast ions and the fusion reactivity profile demonstrating that the Vlasov-Fokker-Planck PINN provides an accurate and efficient means of determining the impact of Knudsen layer yield reduction across a broad range of plasma conditions.

An adjoint formulation of energetic particle confinement in axisymmetric tokamak geometry is derived and evaluated using a physics-informed neural network (PINN). The PINN estimates the mean escape time of energetic ions by solving an inhomogeneous adjoint of the drift kinetic equation with a Lorentz collision operator, yielding predictions of fast ion loss in tokamak geometry due to direct ion orbit loss and collisional transport. To our knowledge, this is the first time a PINN has been used to solve the drift kinetic equation in tokamak geometry, a challenging problem due to the large time scale separation between the rapid transit time of energetic ions and their slow collisional time scale. It is shown that a careful and intentional design of a PINN is able to learn the mean escape time across the majority of the plasma volume, suggesting a path toward constructing a rapid surrogate for use within a broader optimization framework.

An adjoint formulation of energetic particle confinement in axisymmetric geometry is derived and evaluated using a Physics-Informed Neural Network (PINN). The PINN estimates the escape time of energetic ions by solving an inhomogeneous adjoint of the drift kinetic equation with a Lorentz collision operator, yielding predictions of the escape time of fast ions in tokamak geometry due to direct ion orbit loss and collisional transport. This is the first time a PINN has been used to solve the drift kinetic equation in tokamak geometry, a challenging problem due to the large time scale separation present between the rapid transit time of energetic ions, and their slow collision time scale. It is shown that a careful and intentional design of a PINN is able to learn the escape time for the majority of the geometry considered, suggesting a path toward constructing a rapid surrogate for use in a broader optimization framework.

A physics-constrained deep learning surrogate that predicts the exponential ``avalanche'' growth rate of runaway electrons (REs) for a plasma containing partially ionized impurities is developed. Specifically, a physics-informed neural network (PINN) that learns the adjoint of the relativistic Fokker-Planck equation in steady-state is derived, enabling a rapid surrogate of the RE avalanche for a broad range of plasma parameters, motivating a path towards an ML-accelerated integrated description of a tokamak disruption. A steady-state power balance equation together with atomic physics data is embedded directly into the PINN, thus limiting the PINN to train across physically consistent temperatures and charge state distributions. This restricted training domain enables accurate predictions of the PINN while drastically reducing the computational cost of training the model. In addition, a novel closure for the relativistic electron population used when evaluating the secondary source of REs is developed that enables improved accuracy compared to a Rosenbluth-Putvinski source. The avalanche surrogate is verified against Monte Carlo simulations, where it is shown to accurately predict the RE avalanche growth rate across a broad range of plasma parameters encompassing distinct tokamak disruption scenarios.

In a post-thermal-quench plasma, mitigated or unmitigated, the plasma power balance is mostly between collisional or Ohmic heating and plasma radiative cooling. In a plasma of atomic mixture \\(\\{n_\\alpha\\}\\) with \\(\\alpha\\) labeling the atomic species, the power balance sets the plasma temperature, ion charge state distribution \\(\\{n_\\alpha^i\\}\\) with \\(i\\) the charge number, and through the electron temperature \\(T_e\\) and ion charge state distribution \\(\\{n_\\alpha^i\\},\\) the parallel electric field \\(E_\\parallel.\\) Since the threshold electric field for runaway avalanche growth \\(E_{av}\\) is also set by the atomic mixture, ion charge state distribution and its derived quantity, the electron density \\(n_e,\\) the plasma power balance between Ohmic heating and radiative cooling imposes a stringent constraint on the plasma regime for avoiding and minimizing runaways when a fusion-grade tokamak plasma is rapidly terminated.

A physics-informed neural network (PINN) is developed, for the first time, to learn the time-dependent quasi-static magnetohydrodynamic (MHD) equations in axisymmetric tokamak geometry, without any experimental or synthetic data. The initial study considered an ITER-like tokamak and found that a PINN, after careful treatment, was capable of learning the solution to the MHD system and predict a vertically displacing plasma, where general agreement with ground truth simulation was observed. The proof-of-principle demonstration highlights the potential of physics-constrained deep learning to learn complex plasma behavior.

High resolution simulations of incompressible flows have become routine across a range of engineering applications. Despite their routine use, due to the high dimensional parameter space present for most practical applications, a comprehensive exploration of the available parameter space is often impractical. In this work, we demonstrate the ability of physics-constrained deep learning methods to provide an efficient means of exploring high-dimensional parameter spaces with minimal amounts of data from high resolution computational fluid dynamic simulations. As a specific application, we choose the well established problem of a 2D lid driven cavity flow. While giving an extensive treatment of the classic case of a square cavity, we extend the analysis to treat an isosceles trapezoid. In so doing, the number of parameters determining the solution includes not just the Reynolds number, but also two additional parameters characterizing the geometry of the cavity. Thus, together with the \\(\\left( x, y \\right)\\) variation of the flow and pressure in configuration space, the presence of these three parameters results in the solution varying in a 5D space. It is shown that in the absence of data, physics-constrained methods are able to provide an accurate description of the cavity flow in this 5D space up to intermediate values of the Reynolds number, but fails to train for sufficiently high Reynolds numbers. In contrast, using a small quantity of flow data, a single neural network is able to provide an accurate description for a broad range of Reynolds numbers and cavity geometries. Once trained, such a model provides a rapid surrogate that can be used to efficiently explore the 5D space. This 5D surrogate model is subsequently used to identify critical parameter values for the merger and splitting of vortices as the Reynolds number and cavity geometry are varied.