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 towards constructing a rapid surrogate for use within a broader optimisation framework.

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.

Accurate prediction of plasma sheath behavior near material and geometric discontinuities is essential in plasma-based microelectronic manufacturing, where small variations in ion energy and angular distributions can significantly affect semiconductor yield. This work presents a sheath plasma physics-informed neural network (SPPINN) framework for rapid modeling of plasma sheaths in reactive ion etch reactors. The two-dimensional, mesh-free model embeds Poisson's, ion continuity, and ion momentum equations directly into the loss function, ensuring physically constrained solutions. Reactor geometry, including focus ring height, is incorporated as an input parameter, enabling high-dimensional design exploration within a single model. The framework solves ion dynamics and electrostatic potential in a weakly self-consistent manner, where electrons are modeled using a trainable Boltzmann relation based on a prescribed ion source. The model achieves residuals <= 10(-2) for Poisson's equation and <= 10(-3) for continuity and momentum. Predictions are compared against an analogous two-dimensional reference study, showing good agreement in overall sheath structure, with relative L-2 errors of order 10(-2), and limited agreement of ion tilt. Localized discrepancies near sharp geometric transitions arise from differentiable geometry smoothing required for stable training. Ion tilt results were corroborated as physical solutions by a Pearson correlation coefficient >= 0.93, implying the geometry smoothing approximations preserve the dominant physical trends in sheath dynamics while introducing only localized, nonsystematic bias near discontinuities. Once trained, SPPINN generates millisecond-scale predictions across the parametric domain, providing >4 orders-of-magnitude speedup compared to conventional solvers. These results demonstrate SPPINN's viability for rapid, geometry-aware plasma sheath prediction and reactor design exploration.

Despite their ubiquity, the rich physics present in a plasma sheath has inhibited the development of a generally applicable description of this critical region. The present study utilizes a physics-informed neural network (PINN) to evaluate a hierarchy of models of the plasma sheath. Unlike traditional deep learning methods, PINNs use the governing partial differential equations to constrain the predictions of a neural network, and thus do not require any experimental or simulation data to train. In this work, we utilize a PINN to identify the parametric solution to fluid models of different physics fidelity of the plasma sheath. While the offline training time of the PINN is often longer than a traditional solver, once trained, the PINN is able to efficiently predict the sheath profiles across a broad range of parameter regimes, thus yielding an effective surrogate of the plasma sheath.

Hall effect thruster (HET) physics has been studied extensively with fluid,Particle-In-Cell (PIC), direct simulation Monte Carlo (DSMC), and hybrid models,but Physics-Informed Neural Networks (PINNs) remain largely unexploredin electric propulsion. Here we introduce HallPINN, a parametric PINN for Hallthruster physics. The anode mass flow rate and a discharge-current rate enter thenetwork alongside the axial coordinate, so one trained network carries the steady-stateaxial solution over a continuous envelope of operating conditions rather thana single design point. Five steady-state fluid equations are closed by current conservation;current conservation and total mass flux are enforced exactly throughthe output transform, rendering the neutral continuity residual redundant andleaving four residuals in the loss. Training combines 1,500 epochs of first-orderSOAP optimization with 25,000 epochs of the self-scaled Broyden (SSBroyden)quasi-Newton method at a total cost of ≈1,018 s on a single GPU. The quasi-Newton phase drives every governing equation roughly seven orders of magnitudebelow the first-order plateau, to raw mean-squared residuals of 10-14 to 10-11.Training is a one-time offline cost; any operating point is then recovered in a singleforward pass, so the network acts as a surrogate. Benchmarked against published one-dimensional stationary fluid results (1), the network reaches 𝑹2 ≥ 0.994across twelve quantities with relative 𝑳2 errors of 0.96–4.6%, and predicts thrustand specific impulse to within 1.19% and 1.90% while recovering the dischargecurrent, never imposed, to 0.9%.

A physics-constrained deep learning surrogate that predicts the exponential 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.

This work extends the adjoint-deep learning framework for runaway electron (RE) evolution developed in Ref. [C. McDevitt et al., A physics-constrained deep learning treatment of runaway electron dynamics, Submitted to Physics of Plasmas (2024)] to account for large-angle collisions. By incorporating large-angle collisions the framework allows the avalanche of REs to be captured, an essential component to RE dynamics. This extension is accomplished by using a Rosenbluth-Putvinski approximation to estimate the distribution of secondary electrons generated by large-angle collisions. By evolving both the primary and multiple generations of secondary electrons, the present formulation is able to capture both the detailed temporal evolution of a RE population beginning from an arbitrary initial momentum space distribution, along with providing approximations to the saturated growth and decay rates of the RE population. Predictions of the adjoint-deep learning framework are verified against a traditional RE solver, with good agreement present across a broad range of parameters.

An adjoint formulation leveraging a physics-informed neural network (PINN) is employed to advance the density moment of a runaway electron (RE) distribution forward in time. A distinguishing feature of this approach is that once the adjoint problem is solved, its solution can be used to project the RE density forward in time for an arbitrary initial momentum space distribution of REs. Furthermore, by employing a PINN, a parametric solution to the adjoint problem can be learned. Thus, once trained, this adjoint-deep learning framework is able to efficiently project the RE density forward in time across various plasma conditions while still including a fully kinetic description of RE dynamics. As an example application, the temporal evolution of the density of primary electrons is studied, with particular emphasis on evaluating the decay of a RE population when below threshold. Predictions from the adjoint-deep learning framework are found to be in good agreement with a traditional relativistic electron Fokker-Planck solver, for several distinct initial conditions, and across an array of physics parameters. Once trained the PINN thus provides a means of generating RE density time histories with exceptionally low online execution time.

High resolution simulations of incompressible Navier-Stokes 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 two-dimensional 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 (x; y) variation of the flow and pressure in configuration space, the presence of these three parameters results in the solution varying in a five-dimensional 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 five-dimensional 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 for predicting the flow structure and can thus be used to efficiently explore the five dimensional space. This five-dimensional 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.

A reduced kinetic method (RKM) with a first-principle collision operator is introduced in a 1D2V planar geometry and implemented in a computationally inexpensive code to investigate non-local ion heat transport in multi-species plasmas. The RKM successfully reproduces local results for multi-species ion systems and the important features expected to arise due to non-local effects on the heat flux are captured. In addition to this, novel features associated with multi-species, as opposed to single species, case are found. Effects of non-locality on the heat flux are investigated in mass and charge symmetric and asymmetric ion mixtures with temperature, pressure, and concentration gradients. In particular, the enthalpy flux associated with diffusion is found to be insensitive to sharp pressure and concentration gradients, increasing its significance in comparison to the conductive heat flux driven by temperature gradients in non-local scenarios. The RKM code can be used for investigating other kinetic and non-local effects in a broader plasma physics context. Due to its relatively low computational cost it can also serve as a practical non-local ion heat flux closure in hydrodynamic simulations or as a training tool for machine learning surrogates.

A surrogate model of the runaway electron avalanche growth rate in a magnetic fusion plasma is developed. This is accomplished by employing a physics-informed neural network (PINN) to learn the parametric solution of the adjoint to the relativistic Fokker-Planck equation. The resulting PINN is able to evaluate the runaway probability function across a broad range of parameters in the absence of any synthetic or experimental data. This surrogate of the adjoint relativistic Fokker-Planck equation is then used to infer the avalanche growth rate as a function of the electric field, synchrotron radiation and effective charge. Predictions of the avalanche PINN are compared against first principle calculations of the avalanche growth rate with excellent agreement observed across a broad range of parameters.

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.

The exponential growth (avalanching) of runaway electrons (REs) during a tokamak disruption continues to be a large uncertainty in RE modeling. The present work investigates the impact of tokamak geometry on the efficiency of the avalanche mechanism across a broad range of disruption scenarios. It is found that the parameter ν*,crit, describing the collisionality at the critical energy to run away, delineates how toroidal geometry impacts RE formation. In particular, utilizing a reduced but self-consistent description of plasma power balance, it is shown that for a high-density deuterium-dominated plasma, ν*,crit is robustly less than one, resulting in a substantial decrease in the efficiency of the RE avalanche compared to predictions from slab geometry. In contrast, for plasmas containing a substantial quantity of neon or argon, ν*,crit≳1, no reduction of the avalanche is observed due to toroidal geometry. This sharp contrast in the impact of low- vs high-Z material results primarily from the relatively strong radiative cooling from high-Z impurities, enabling the plasma to be radiatively pinned at low temperatures and thus large electric fields, even for modest quantities of high-Z material.

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, the 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 the 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.

A challenging aspect of the description of a tokamak disruption is evaluating the hot tail runaway electron (RE) seed that emerges during the thermal quench. This problem is made challenging due to the requirement of describing a strongly non-thermal electron distribution, together with the need to incorporate a diverse range of multiphysics processes including magnetohydrodynamic instabilities, impurity transport, and radiative losses. The present work develops a physics-informed neural network (PINN) tailored to the solution of the hot tail seed during an idealized axisymmetric thermal quench. Here, a PINN is developed to identify solutions to the adjoint relativistic Fokker-Planck equation in the presence of a rapid quench of the plasma's thermal energy. It is shown that the PINN is able to accurately predict the hot tail seed across a range of parameters including the thermal quench time scale, initial plasma temperature, and local current density, in the absence of experimental or simulation data. The hot tail PINN is verified by comparison with a direct Monte Carlo solution, with excellent agreement found across a broad range of thermal quench conditions.

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 {nα} with α labeling the atomic species, the power balance sets the plasma temperature, ion charge state distribution {nαi} with i the charge number, and through the electron temperature Te and ion charge state distribution {nαi}, the parallel electric field E∥. Since the threshold electric field for runaway avalanche growth Eav is also set by the atomic mixture, ion charge state distribution and its derived quantity, the electron density ne, 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.

Benign termination of mega-ampere (MA) level runaway current has been convincingly demonstrated in recent JET and DIII-D experiments, establishing it as a leading candidate for runaway mitigation on ITER. This comes in the form of a runaway flush by parallel streaming loss along stochastic magnetic field lines formed by global magnetohydrodynamic instabilities, which are found to correlate with a low-Z injection that purges the high-Z impurities from a post-thermal-quench plasma. Here, we show the competing physics that govern the postflush reconstitution of the runaway current in an ITER-like reactor where significantly higher current is expected. The trapped "runaways" are found to dominate the seeding for runaway reconstitution, and the incomplete purge of high-Z impurities helps drain the seed but produces a more efficient avalanche, two of which compete to produce a 2-3 MA step in current drop before runaway reconstitution of the plasma current.

Minority relativistic electron populations can occur in a range of complex plasmas. Of specific interest is when runaway electrons form among the presence of high-atomic-number ion species in a tokamak plasma discharge. It has been recently demonstrated that ion charge state distributions and radiation losses at low bulk electron temperatures can be dominated by relativistic electrons, even though their density is orders of magnitude lower. This was attributed to the relativistic enhancement of electron impact inelastic cross sections. In this work, we provide a closer inspection of the atomic physics underpinning this effect. We also demonstrate the consequences of runaway enhanced scattering on post-disruption tokamak fusion discharges with neon and argon impurities present. Effects on charge state distributions, radiation and spectral characteristics, and reduced-order modeling considerations are discussed.