Inverse Modeling for Laser Pulse Shape Design in Inertial Confinement Fusion

Ricardo Luna Gutierrez; Vineet Gundecha; Rahman Ejaz; Varchas Gopalaswamy; Riccardo Betti; Sahand Ghorbanpour; Aarne Lees; Soumyendu Sarkar

cs.AIOct 22, 2026

Inverse Modeling for Laser Pulse Shape Design in Inertial Confinement Fusion

Authors: Ricardo Luna Gutierrez, Vineet Gundecha, Rahman Ejaz, Varchas Gopalaswamy, Riccardo Betti, Sahand Ghorbanpour, Aarne Lees, Soumyendu Sarkar

Organizations: Hewlett Packard Enterprise · University of Rochester

Abstract

The achievement of practical fusion energy remains one of the most pressing unsolved scientific challenges, with immense implications for carbon-free power. A critical factor in the success of Inertial Confinement Fusion (ICF) is the design of a Laser Pulse Shape (LP) that can optimally drive implosions under stringent physical constraints. Traditional LP design relies on computationally expensive simulations and manual iterative refinement. We introduce the ICF Laser Pulse Shape Design System (LPDS), a generative inverse modeling framework that maps desired outcomes and target pellet configurations directly to optimized LPs. Crucially, we design a multi-objective loss function to ensure the generated LPs satisfy fundamental physical constraints and experimental requirements. Furthermore, we present constraint-conditioning, inpainting, and gradient-based LP editing mechanisms for fine-grained control over specific pulse characteristics during generation. Moreover, we study the transfer of our framework to a limited set of real experimental shots. Our method establishes a data-driven inverse design framework for LP design in ICF, contributing to the advancement of practical and sustainable fusion energy.

Figures & tables

Figure 1: Inverse design workflow. (1) Fit an outcome surrogate to simulations. (2) Train the inverse model using examples, outcome agreement, and constraints; freeze the surrogate. (3) Generate designs from desired outcomes and system properties without simulator or surrogate calls.

LPDS variant	Diversity $\uparrow$	$\mathbf{m}$ Error $\downarrow$	Reconstruction $\downarrow$	Energy Cons. $\downarrow$	LILAC Error $\downarrow$
		(%)	( $\times 10^{-3}$ )	(%)	(%)
Diff	$0.64\pm 0.077$	$1.93\pm 0.046$	$5.6\pm 0.20$	$1.59\pm 0.14$	$2.07$
LSTM	–	$1.65\pm 0.09$	$0.1\pm 0.05$	$0.66\pm 0.07$	$2.01$
Transformer	–	$1.94\pm 0.11$	$0.8\pm 0.08$	$0.95\pm 0.10$	–
LSTM ${}_{\text{Gaussian}}$	$0.42\pm 0.02$	$1.89\pm 0.01$	$0.5\pm 0.02$	$0.58\pm 0.004$	–
LSTM ${}_{\text{MixtureOfGaussian}}$	$0.56\pm 0.10$	$1.95\pm 0.09$	$0.6\pm 0.08$	$1.58\pm 0.006$	–

Table 1: ICF model performance.

\pm

denotes standard deviation over seeds and training runs. For the predictive autoregressive models (

\mathbf{LPDS}_{\text{LSTM}}

and

\mathbf{LPDS}_{\text{Transformer}}

), diversity is not defined because they are deterministic. The LILAC-error column reports true simulator validation rather than surrogate evaluation and is available only for the strongest models because simulator access is limited. The ablation experiments for the loss terms were performed for a subset of the models due to computational constraints.

Figure 2: Overview of LPDS capabilities. Panel (a) shows diverse conditional generation of valid pulse shapes, panel (b) shows constraint-based editing of specific pulse attributes, panel (c) shows inpainting of partially specified pulses, and panel (d) shows outcome-driven optimization followed by simulator validation.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: ICF-specific LPDS architecture. The inverse model $G_{\theta}$ generates a laser pulse $\mathbf{l}^{\prime}$ from embedded pellet parameters $\mathbf{p}$ and desired implosion outcomes $\mathbf{m}$ . Training combines pulse reconstruction, surrogate-predicted outcome agreement, and physics-informed penalties. The dashed branch adds optional pulse-shape constraints $\mathbf{c}$ through a pulse-to-parameter model $\mathcal{M}$ .

Figure 4: The LP is typically characterized by 12 parameters, some of which are annotated here.

Transformer layers	4
Attention heads	4
Feed forward dimension	2048
Embedding layers	2
Embedding dimension	1024
Dropout	0.1
Learning rate	$5e^{-5}$

Appendix

Table 2: Network Hyperparameters

\epsilon_{\theta}

used in diffusion model.

Num of layers	4
Number of hidden units	512
Learning rate	$1e^{-5}$
Batch size	128

Appendix

Table 3: Network Hyperparameters used in the LSTM autoregressive model.

$\lambda_{\mathcal{S}}$	0.5
$\lambda_{\mathcal{P}}$	0.25
$\lambda_{\mathcal{L}}$	0.25

Appendix

Table 4: Hyperparameters used for our inverse-model objective

\mathcal{O}(\theta)

Num. of layers	4
Num. of hidden units	256
Activation function	ReLU

Appendix

Table 5: Hyperparameters used for the LILAC surrogate

\mathbf{S_{\phi}}

Surrogate	Test MAPE (%) $\downarrow$
MLP (used in LPDS)	$\mathbf{1.40}$
FNO (1D, 32 modes, width 64)	1.43
Physics-informed MLP ( $\tanh$ , energy-conservation residual)	2.31

Appendix

Table 6: Comparison of surrogate architectures for approximating the LILAC outcomes

\mathbf{m}

LPDS variant / Baseline	Diversity $\uparrow$	$\mathbf{m}$ Error $\downarrow$	Reconstruction $\downarrow$	Energy Cons. $\downarrow$
		(%)	( $\times 10^{-3}$ )	(%)
LPDS Diff	$0.64\pm 0.077$	$1.93\pm 0.046$	$5.6\pm 0.20$	$1.59\pm 0.14$
LPDS LSTM	–	$1.65\pm 0.09$	$0.1\pm 0.05$	$0.66\pm 0.07$
VAE	$0.16\pm 0.06$	$4.97\pm 0.24$	$1.0\pm 0.10$	$0.61\pm 0.04$
GAN	$0.68\pm 0.09$	$9.88\pm 0.73$	$4.0\pm 0.20$	$0.78\pm 0.09$
TabSyn	$0.66\pm 0.13$	$15.80\pm 0.80$	$23.0\pm 0.20$	$4.76\pm 0.07$

Appendix

Table 7: Comparison of LPDS with off-the-shelf conditional generators and inverse-problem methods on the simulation test set.

\pm

denotes standard deviation over seeds and training runs. Diversity is not defined for deterministic methods.

Panel of Figure 5 — Figure 5: LPDS variants compared with off-the-shelf baselines on each metric. Error bars denote standard deviation over seeds and training runs. Diversity is omitted for deterministic models.

Panel of Figure 6 — Figure 6: Ablation of the surrogate loss $\mathcal{S}$ and the physics-informed loss $\mathcal{P}$ (Table 1 ). Removing $\mathcal{S}$ more than doubles the $\mathbf{m}$ error of all three models, while removing $\mathcal{P}$ roughly doubles the energy violation of $\mathbf{LPDS}_{\text{Diff}}$ .

Panel of Figure 7 — Figure 7: Fidelity–Diversity analysis: (a) trade-off curve and (b) measured metrics across $\lambda_{\mathcal{L}}$ and $\lambda_{\mathcal{S}}$ settings.

Approach	Maximum Power Cap	Maximum Cumulative Energy
LSTM	0 $\%$	0 $\%$
Transformer	0 $\%$	0 $\%$
Diffusion	0 $\%$	0 $\%$
LSTM ${}_{\text{Gaussian}}$	0 $\%$	0 $\%$
LSTM ${}_{\text{MixtureOfGaussian}}$	0 $\%$	0 $\%$
LSTM ${}_{\text{Categorical}}$	0 $\%$	0 $\%$

Appendix

Table 8: Violation rates of the benchmarked approaches for LP generation.

Setting	$\mathbf{m}$ Error (%) $\downarrow$
Random split, $\mathbf{LPDS}_{\text{Diff}}$ (Table 1 )	1.93
Pellet-stratified split, ground-truth pulses (surrogate floor)	1.53
Pellet-stratified split, $\mathbf{LPDS}_{\text{Diff}}$	2.02

Appendix

Table 9: Generalization of

\mathbf{LPDS}_{\text{Diff}}

to held-out pellet clusters.

Approach	Avg. Training Time (h)	Throughput (sample/s)	Latency (ms)
LSTM	10.21 $\pm$ 0.23	16666	26.8
Transformer	8.42 $\pm$ 0.45	34666	18.3
Diffusion	29.58 $\pm$ 1.4	1939	528
LSTM ${}_{\text{Gaussian}}$	12.21 $\pm$ 0.66	16666	26.8
LSTM ${}_{\text{MixtureOfGaussian}}$	13.43 $\pm$ 0.75	16666	26.8
LSTM ${}_{\text{Categorical}}$	12.13 $\pm$ 0.8	16666	26.8

Appendix

Table 10: Computational efficiency of the different approaches used for ICF LP generation.

Population	Mean novelty	95th percentile	Fraction beyond test 95th percentile
Training (nearest other training pulse)	0.7125	1.0807	–
Test	0.7125	1.0814	5.00%
$\mathbf{LPDS}_{\text{Diff}}$ (held-out pellet cluster)	0.7048	–	5.04%

Appendix

Table 11: Novelty of generated pulses relative to the training set, compared with held-out test pulses.

Panel of Figure 10 — Figure 10: LPs generated by predictive autoregressive models.

Panel of Figure 11 — Figure 11: LPs generated by generative autoregressive model

Panel of Figure 12 — Figure 12: Examples of LPs generated by $\mathbf{LPDS}_{\text{Diff}}$ with the outcome-optimization procedure described in Section 5.5 .

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Offline-trained surrogates for Inertial Confinement Fusion (ICF) suffer a well-known failure mode that iterative optimizers drive inputs into out-of-distribution (OOD) regions where predictions become unreliable. Here we present Co4ICF, a co-evolving framework that couples a physics-informed surrogate with a PPO-based pulse optimizer. The surrogate is iteratively fine-tuned on policy-induced trajectories, correcting extrapolation errors as the optimizer shifts the input distribution; the optimizer queries this evolving surrogate as a fast environment. In the 1D MULTI environment, Co4ICF achieves 146.1% normalized yield based on current laser design baseline; as a post-hoc cross-fidelity check, the optimized pulse further attains 246.9% normalized yield when directly evaluated in 2D-MULTI without any 2D training or fine-tuning. Budget-matched ablations support that the gains are not explained solely by additional simulation data and are consistent with the co-evolving mechanism playing a key role. We release a large-scale MULTI-IFE simulation dataset to support future benchmarking.

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