Learning gradient flow: Using equation discovery to accelerate engineering optimization

doi:10.1016/j.cma.2026.119099

Learning gradient flow: Using equation discovery to accelerate engineering optimization

2026-05-01

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DOI：10.1016/j.cma.2026.119099

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In this work, we investigate the use of data-driven equation discovery for dynamical systems to model and forecast continuous-time dynamics of unconstrained optimization problems. To reduce expensive evaluations of the objective function and its gradient, we leverage trajectory data on the optimization variables to learn the continuous-time dynamics associated with gradient descent, Newton's method, and ADAM optimization. The discovered gradient flows are then solved as a surrogate for the original optimization problem. To this end, we introduce the Learned Gradient Flow (LGF) optimizer, which is equipped to build surrogate models of variable polynomial order in full-or reduced-dimensional spaces at user-defined intervals in the optimization process. We demonstrate the efficacy of this approach on several standard problems from engineering mechanics and scientific machine learning, including inverse problems, structural topology optimization, and two numerical solutions of boundary value problems with different discretizations. Our results suggest that the learned gradient flows can significantly expedite convergence by capturing critical features of the optimization trajectory while reducing expensive evaluations of the objective and its gradient.

Keywords:

Equation discovery

Data-driven modeling

Gradient flow

Surrogate models

Optimization

Learning gradient flow: Using equation discovery to accelerate engineering optimization

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