Causal Multi-fidelity Surrogate Forward and Inverse Models for ICF Implosions

TL;DR

The paper tackles inverse design and parameter inference in inertial confinement fusion by introducing a causal, multi-fidelity surrogate that embeds a reduced-order ODE for the DT shell interface and learns both forward and inverse maps from radiation drive histories. The forward model combines a low-fidelity controller network with a high-fidelity residual network to achieve HF-accurate interface trajectories using limited HF data, while enforcing physical causality and a consistent forward–inverse cycle. The inverse analysis includes dense-time and sparse-time drive reconstruction from observed interface dynamics, using PCA to compress drives and a differentiable soft-snapshot selection mechanism for diagnostic-limited scenarios. Together, the framework demonstrates accurate forward simulations, cycle-consistent drive estimation, and informative sparse-time sampling, offering a scalable approach to accelerated discovery and design in high-energy-density systems with strong physical inductive biases and causality.

Abstract

Continued progress in inertial confinement fusion (ICF) requires solving inverse problems relating experimental observations to simulation input parameters, followed by design optimization. However, such high-dimensional dynamic PDE-constrained optimization problems are extremely challenging or even intractable. It has been recently shown that inverse problems can be solved by only considering certain robust features. Here we consider the ICF capsule's deuterium-tritium (DT) interface, and construct a causal, dynamic, multifidelity reduced-order surrogate that maps from a time-dependent radiation temperature drive to the interface's radius and velocity dynamics. The surrogate targets an ODE embedding of DT interface dynamics, and is constructed by learning a controller for a base analytical model using low- and high-fidelity simulation training data with respect to radiation energy group structure. After demonstrating excellent accuracy of the surrogate interface model, we use machine learning (ML) models with surrogate-generated data to solve inverse problems optimizing radiation temperature drive to reproduce observed interface dynamics. For sparse snapshots in time, the ML model further characterizes the most informative times at which to sample dynamics. Altogether we demonstrate how operator learning, causal architectures, and physical inductive bias can be integrated to accelerate discovery, design, and diagnostics in high-energy-density systems.

Figures (10)

• Figure 1: (a) Implosion configuration and (b) ensemble of drives.
• Figure 2: (Top Row) Reference data versus controlled solution $\tilde{{\mathbf{x}}}(t;\tilde{P})=(\tilde{R}_i(t),\tilde{V}_i(t))$ for radius in $[mm]$ (left) and velocity in $[mm/ns]$ (middle) with learned controller $\tilde{P}$in $[PW]$ (right) corresponding to maximum error in the $L_\infty$ norm, which is $\approx 10 \,[\mu m/ns]$ at $t\approx 6.7 \,[ns]$ in the LF velocity. The corresponding HF controller and solutions are also plotted. (Bottom Row) Controller solution error distributions (with denoted $5^\text{th}$, $50^\text{th}$, and $95^\text{th}$ percentiles) for radius in $[mm]$ (red) and velocity in $[mm/ns]$ (blue) computed via $L_\infty$ (left), $L_1$ (middle), and relative (nondimensional)$L_1$ metrics. Note, the difference between LF and HF errors are not statistically significant, and therefore their errors have been aggregated.
• Figure 3: (Left) Worst-case LF test-set predictions (in $L_\infty$) for radius $\hat{\mathbf{R}}_i^{\text{LF}}$ and velocity $\hat{\mathbf{V}}_i^{\text{LF}}$, which come from two different test samples. They are computed by numerically integrating \ref{['eq:bk-control']}, using ${\mathcal{F}}_\text{LF}$ predictions $\hat{\mathbf p}^\text{LF}$ for the controller coefficients $\mathbf p$. Test error distributions for radius in $[mm]$ (red) and velocity in $[mm/ns]$ (blue), with denoted $5^\text{th}$, $50^\text{th}$, and $95^\text{th}$ percentiles, computed via $L_\infty$ (middle) and relative (nondimensional)$L_1$ (right) metrics.
• Figure 4: (Left) Worst-case HF test-set predictions (in $L_\infty$) for radius $\hat{R}_i^{\text{HF}}$ and velocity $\hat{V}_i^{\text{HF}}$, which come from two different test samples. They are computed by numerically integrating \ref{['eq:bk-control']} using ${\mathcal{F}}_\text{HF}$ predictions for $\hat{\mathbf p}^\text{HF}$ the controller coefficients $\mathbf p$. Test error distributions for radius in $[mm]$ (red) and velocity in $[mm/ns]$ (blue), with denoted $5^\text{th}$, $50^\text{th}$, and $95^\text{th}$ percentiles, computed via $L_\infty$ (middle) and relative (nondimensional)$L_1$ (right) metrics.
• Figure 5: (Left) Median- and worst-case drive test-set predictions (in $L_\infty$) for inverse model $\mathcal{I}_\text{D}$. (Middle) Test error distributions via $L_\infty$in $[eV]$ (pink) and relative $L_1$(nondimensional) (green) metrics (with denoted $5^\text{th}$, $50^\text{th}$, and $95^\text{th}$ percentiles).

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4: Global time selection
• Remark 5