Deep Legendre Transform

We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions, a fundamental operation in convex analysis with various applications in different fields such as optimization, control theory, physics and economics. While traditional numerical methods suffer from the curse of dimensionality and become computationally intractable in high dimensions, more recent neural network-based approaches scale better, but have mostly been studied with the aim of solving optimal transport problems and require the solution of complicated optimization or max-min problems. Using an implicit Fenchel formulation of convex conjugation, our approach facilitates an efficient gradient-based framework for the minimization of approximation errors and, as a byproduct, also provides a posteriori estimates of the approximation accuracy. Numerical experiments demonstrate our method's ability to deliver accurate results across different high-dimensional examples. Moreover, by employing symbolic regression with Kolmogorov-Arnold networks, it is able to obtain the exact convex conjugates of specific convex functions. Code is available at https://github.com/lexmar07/Deep-Legendre-Transform * Corresponding author 1 The Legendre transform-a fundamental operation for switching between dual formulations and variables-has broad utility across thermodynamics, mechanics, optimization, and economics. Key applications include: switching between thermodynamic potentials by exchanging extensive variables (entropy, volume) for intensive conjugates (temperature, pressure) in physics; constructing Moreau envelopes in variational analysis; deriving indirect utility and profit functions in economics; and computing convex potentials in optimal transport. Recent work has further demonstrated applications in approximate dynamic programming Sharifi Kolarijani and Mohajerin Esfahani [2023] and self-concordant smoothing Adeoye and Bemporad [2023]. This paper introduces the Deep Legendre Transform algorithm, establishes its theoretical foundations, and demonstrates its application to optimal-control problems governed by Hamilton-Jacobi equations.

39th Conference on Neural Information Processing Systems (NeurIPS 2025).