# Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics

arXiv: https://arxiv.org/abs/2605.20639 · HTML: https://mlsanity.com/paper/2605.20639/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 71 references that are in the map.
- First posted on arXiv 20 May 2026.

## Builds on (most cited first, 23 of 23)

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques](https://mlsanity.com/paper/2107.11460/) (2021)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population](https://mlsanity.com/paper/2204.14141/) (2022)
- [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)
- [Model-Based Derivative-Free Optimization Methods and Software](https://mlsanity.com/paper/2210.12018/) (2022)
- [A practical identifiability criterion leveraging weak-form parameter estimation](https://mlsanity.com/paper/2506.17373/) (2025)
- [Weak Form Scientific Machine Learning: Test Function Construction for System Identification](https://mlsanity.com/paper/2507.03206/) (2025)
- [Learning Physically Interpretable Atmospheric Models from Data with WSINDy](https://mlsanity.com/paper/2501.00738/) (2025)
- [Bias and Coverage Properties of the WENDy-IRLS Algorithm](https://mlsanity.com/paper/2510.03365/) (2025)
- [Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movement](https://mlsanity.com/paper/2510.07786/) (2025)
- [An Adjoint Method for a High-Order Discretization of Deforming Domain Conservation Laws for Optimization of Flow Problems](https://mlsanity.com/paper/1512.04619/) (2015)
- [Sequential decoder training for improved latent space dynamics identification](https://mlsanity.com/paper/2510.03535/) (2025)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)

## Abstract (arXiv)

Optimization problems constrained by high-dimensional, time-dependent partial differential equations require repeated forward and sensitivity solves, making high-fidelity optimization computationally prohibitive in many-query design and control settings. We present a weak-form latent-space reduced-order modeling framework for accelerating gradient-based PDE-constrained optimization. The proposed approach builds on Weak-form Latent Space Dynamics Identification (WLaSDI), which compresses high-dimensional solution trajectories into a low-dimensional latent representation and identifies parametric latent dynamics using weak-form system identification. By avoiding explicit numerical differentiation of training trajectories, the weak-form improves robustness to noisy data and yields more reliable surrogate dynamics for optimization. We formulate the resulting reduced PDE-constrained optimization problem and derive both direct-sensitivity and adjoint-based gradient expressions for the learned latent dynamics, enabling scalable gradient evaluation with respect to design parameters. The framework is demonstrated on three time-dependent benchmark problems: thermal radiative transfer for optimal hohlraum design, the two-stream instability Vlasov-Poisson system, and the inviscid Burgers equation. Across these examples, WLaSDI produces accurate optimal designs, remains robust under noisy training data, and delivers substantial computational savings, including speedups of up to five orders of magnitude relative to full-order optimization. These results demonstrate that weak-form latent dynamics provide an efficient and noise-robust surrogate foundation for gradient-based optimization of complex time-dependent PDE systems.