[Paper Review] Optimal control gradient precision trade-offs: application to fast generation of DeepControl libraries for MRI

Mads Sloth Vinding, David L. Goodwin+2 more|arXiv (Cornell University)|Jul 2, 2021

Advanced NMR Techniques and Applications59 references9 citations

TL;DR

This paper proposes a midpoint first-order gradient approximation method for quantum optimal control in MRI pulse design, significantly accelerating training library generation for DeepControl neural networks. The method achieves up to fivefold speedup over machine-precision gradients while maintaining sufficient accuracy for practical MRI applications, enabling real-time patient-specific pulse computation via deep learning.

ABSTRACT

We have recently demonstrated supervised deep learning methods for rapid generation of radiofrequency pulses in magnetic resonance imaging (https://doi.org/10.1002/mrm.27740, https://doi.org/10.1002/mrm.28667). Unlike the previous iterative optimization approaches, deep learning methods generate a pulse using a fixed number of floating-point operations - this is important in MRI, where patient-specific pulses preferably must be produced in real time. However, deep learning requires vast training libraries, which must be generated using the traditional methods, e.g. iterative quantum optimal control methods. Those methods are usually variations of gradient descent, and the calculation of the fidelity gradient of the performance metric with respect to the pulse waveform can be the most numerically intensive step. In this communication, we explore various ways in which the calculation of fidelity gradients in quantum optimal control theory may be accelerated. Four optimization avenues are explored: truncated commutator series expansions at zeroth and first order, a novel midpoint truncation scheme at first order, and the exact complex-step method. For the spin systems relevant to MRI, the first-order truncation is found to be sufficiently accurate, but also up to five times faster than the machine precision gradient. This makes the generation of training databases for the machine learning methods considerably more realistic.

Motivation & Objective

• To address the computational bottleneck in generating large-scale training libraries for DeepControl, a deep learning framework for real-time MRI RF pulse generation.
• To explore trade-offs between gradient precision and computational efficiency in quantum optimal control for MRI.
• To develop and evaluate approximate gradient methods that reduce runtime without sacrificing pulse fidelity.
• To enable faster, scalable generation of high-quality RF pulse libraries for deep learning-based MRI pulse design.

Proposed method

• Proposes a novel midpoint first-order gradient approximation method for the GRAPE algorithm, using a time-symmetric midpoint evaluation of the control field.
• Compares four gradient computation methods: exact complex-step, standard zeroth- and first-order approximations, and the proposed midpoint first-order method.
• Uses piecewise-constant pulse discretization with time steps ∆t = 10 µs and applies the matrix exponential to compute time-propagators U(p)n = e^{Ω(p)n∆t}.
• Evaluates gradient accuracy and convergence speed across three target categories: BW (bandwidth), GrBW (gradient-echo bandwidth), and Gr (gradient-echo).
• Employs NRMSE (normalized root mean square error) to compare approximate gradients against the exact complex-step gradient as a reference.
• Tests the methods on 2D RF pulses targeting a nominal 30° flip angle, with convergence assessed via fidelity and pulse shape accuracy.

Experimental results

Research questions

• RQ1 Can a first-order gradient approximation with midpoint evaluation outperform standard first-order and zeroth-order approximations in speed and accuracy for MRI pulse optimization?
• RQ2 To what extent does the midpoint gradient method reduce computational cost while preserving pulse fidelity compared to exact gradients?
• RQ3 How do different gradient approximation levels affect convergence rate and final pulse quality in the context of DeepControl library generation?
• RQ4 Is the midpoint method robust across diverse pulse design targets (e.g., BW, GrBW, Gr) and pulse types?

Key findings

• The midpoint first-order gradient method achieves up to fivefold speedup over machine-precision complex-step gradients while maintaining sufficient accuracy for MRI applications.
• The method’s accuracy is comparable to the standard zeroth-order gradient with a halved time step (5 µs vs. 10 µs), indicating it effectively reduces time-step sensitivity.
• The first-order approximation (including midpoint) is sufficiently accurate for MRI, as the resulting pulses achieve high fidelity and are suitable for training DeepControl networks.
• NRMSE histograms show that the midpoint method consistently outperforms standard first-order and zeroth-order approximations across all target categories (BW, GrBW, Gr).
• The best-case NRMSE for the midpoint method was below 0.005, with worst-case differences in flip angle maps remaining within ±20°, indicating acceptable accuracy.
• The study confirms that approximate gradients can be used effectively in place of exact gradients, especially when the primary goal is fast library generation rather than ultra-high precision.