publications

2026

1. OCsPOD

   Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework

   Tobias Breiten, Shubhaditya Burela, and Philipp Schulze

   arXiv, 2026

   Abs Bib HTML

   Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing low-dimensional representations of high-fidelity transport-dominated phenomena. In this work, a reduced-order model is constructed first, followed by the optimization of the reduced system. We consider a 1D linear advection equation problem and prove existence and uniqueness of solutions for the reduced-order model as well as the existence of an optimal control. Moreover, we compare the computational performance of the shifted POD method against the standard POD.

   @article{BrBuSc,
     title = {Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework},
     author = {Breiten, Tobias and Burela, Shubhaditya and Schulze, Philipp},
     journal = {arXiv},
     year = {2026},
     doi = {https://doi.org/10.48550/arXiv.2603.28442},
   }

2025

1. sPOD-NN

   Parametric model order reduction for a wildland fire model via the shifted POD based deep learning method

   Shubhaditya Burela, Philipp Krah, and Julius Reiss

   Advances in Computational Mathematics , 2025

   Abs Bib HTML

   Parametric model order reduction techniques often struggle to accurately representtransport-dominated phenomena due to a slowly decaying Kolmogorov n-width. Toaddress this challenge, we propose a non-intrusive, data-driven methodology thatcombines the shifted proper orthogonal decomposition (POD) with deep learning.Specifically, the shifted POD technique is utilized to derive a high-fidelity, low-dimensional model of the flow, which is subsequently utilized as input to a deeplearning framework to forecast the flow dynamics under various temporal and param-eter conditions. The efficacy of the proposed approach is demonstrated through theanalysis of one- and two-dimensional wildland fire models with varying reaction rates,and its error is compared with the error of other similar methods. The results indicatethat the proposed approach yields reliable results within the percent range, while alsoenabling rapid prediction of system states within seconds.

   @article{BuKrRe23,
     title = {Parametric model order reduction for a wildland fire model via the shifted POD based deep learning method},
     author = {Burela, Shubhaditya and Krah, Philipp and Reiss, Julius},
     journal = {Advances in Computational Mathematics },
     year = {2025},
     doi = {https://doi.org/10.1007/s10444-025-10220-4},
   }

2024

1. NsPOD

   Automated transport separation using the neural shifted proper orthogonal decomposition

   Beata Zorawski, Shubhaditya Burela, Philipp Krah, and 2 more authors

   9th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), 2024

   Abs Bib HTML

   This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is difficult or even impossible to obtain, limiting the applicability and benefits of the sPOD. To address this issue, our approach estimates both the transport and co-moving fields simultaneously using neural networks. This is achieved by training two sub-networks dedicated to learning the transports and the co-moving fields, respectively. Applications to synthetic data and a wildland fire model illustrate the capabilities and efficiency of this neural sPOD approach, demonstrating its ability to separate the different fields effectively.

   @article{ZoBuKrMaSc24,
     title = {Automated transport separation using the neural shifted proper orthogonal decomposition},
     author = {Zorawski, Beata and Burela, Shubhaditya and Krah, Philipp and Marmin, Arthur and Schneider, Kai},
     journal = {9th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS)},
     year = {2024},
     doi = {10.23967/eccomas.2024.066},
   }

2023

1. HCB

   Hessian chain bracketing

   Uwe Naumann, and Shubhaditya Burela

   Journal of Combinatorics, 2023

   Abs Bib HTML

   Second derivatives of mathematical models for real-world phenomena are fundamental ingredients of a wide range of numerical simulation methods including parameter sensitivity analysis, uncertainty quantification, nonlinear optimization and model calibration. The evaluation of such Hessians often dominates the overall computational effort. The combinatorial Hessian Accumulation problem aiming to minimize the number of floating-point operations required for the computation of a Hessian turns out to be NP‑complete. We propose a dynamic programming formulation for the solution of Hessian Accumulation over a sub-search space. This approach yields improvements by factors of ten and higher over the state of the art based on second-order tangent and adjoint algorithmic differentiation.

   @article{NaBu23,
     title = {Hessian chain bracketing},
     author = {Naumann, Uwe and Burela, Shubhaditya},
     journal = {Journal of Combinatorics},
     volume = {14(2023)},
     number = {4},
     pages = {539--558},
     year = {2023},
     publisher = {International Press of Boston},
     doi = {https://dx.doi.org/10.4310/JOC.2023.v14.n4.a7},
   }