# End-to-End Learning Framework for Solving Non-Markovian Optimal Control

arXiv: https://arxiv.org/abs/2502.04649 · HTML: https://mlsanity.com/paper/2502.04649/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 51 references that are in the map.
- First posted on arXiv 7 Feb 2025.

## Builds on (most cited first, 15 of 15)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [The Gap Between Model-Based and Model-Free Methods on the Linear Quadratic Regulator: An Asymptotic Viewpoint](https://mlsanity.com/paper/1812.03565/) (2018)
- [Deep State Space Models for Nonlinear System Identification](https://mlsanity.com/paper/2003.14162/) (2020)
- [Deep Energy-Based NARX Models](https://mlsanity.com/paper/2012.04136/) (2020)
- [A novel Deep Neural Network architecture for non-linear system identification](https://mlsanity.com/paper/2106.03078/) (2021)
- [Simultaneous System Identification and Model Predictive Control with No Dynamic Regret](https://mlsanity.com/paper/2407.04143/) (2024)
- [A Sampling Complexity-aware Framework for Discrete-time Fractional-Order Dynamical System Identification](https://mlsanity.com/paper/2501.17499/) (2025)
- [On Learning Discrete-Time Fractional-Order Dynamical Systems](https://mlsanity.com/paper/2103.14975/) (2021)

## Built on by (most cited first, 2 of 2)

- [Cochain Perspectives on Temporal-Difference Signals for Learning Beyond Markov Dynamics](https://mlsanity.com/paper/2602.06939/) (2026)
- [Learning Fractional-Order Dynamics from a Single Trajectory](https://mlsanity.com/paper/2609.18127/) (2026)

## Abstract (arXiv)

Integer-order calculus often falls short in capturing the long-range dependencies and memory effects found in many real-world processes. Fractional calculus addresses these gaps via fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control due to the lack of standard control methodologies. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing an innovative system identification method control strategy for FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving a theoretical analysis of sample complexity to quantify the number of samples required for accurate optimal control in complex real-world problems. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control.