# Inverse Models for Estimating the Initial Condition of Spatio-Temporal Advection-Diffusion Processes

arXiv: https://arxiv.org/abs/2302.04134 · HTML: https://mlsanity.com/paper/2302.04134/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 8 Feb 2023.

## Builds on (most cited first, 7 of 7)

- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Active Learning for Deep Gaussian Process Surrogates](https://mlsanity.com/paper/2012.08015/) (2020)
- [Gaussian Process Assisted Active Learning of Physical Laws](https://mlsanity.com/paper/1910.03120/) (2019)
- [Fast and Flexible ADMM Algorithms for Trend Filtering](https://mlsanity.com/paper/1406.2082/) (2014)
- [Distance-distributed design for Gaussian process surrogates](https://mlsanity.com/paper/1812.02794/) (2018)
- [Development of hp-inverse model by using generalized polynomial chaos](https://mlsanity.com/paper/1801.03009/) (2018)
- [Statistical Treatment of Inverse Problems Constrained by Differential Equations-Based Models with Stochastic Terms](https://mlsanity.com/paper/1810.08557/) (2018)

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

- [Geometry-aware Active Learning of Spatiotemporal Dynamic Systems](https://mlsanity.com/paper/2504.19012/) (2025)
- [P-K-GCN: Physics-augmented Koopman-enhanced Graph Convolutional Network for Deep Spatiotemporal Super-resolution](https://mlsanity.com/paper/2606.19303/) (2026)

## Abstract (arXiv)

Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Three spatial sampling schemes are considered, including irregular, non-uniform and shifted uniform sampling. The irregular sampling scheme is the general scenario, while computationally efficient solutions are available in the spectral domain for non-uniform and shifted uniform sampling. For each sampling scheme, the inverse problem is formulated as a regularized convex optimization problem that minimizes the distance between forward model outputs and observations. The optimization problem is solved by the Alternating Direction Method of Multipliers algorithm, which also handles the situation when a linear inequality constraint (e.g., non-negativity) is imposed on the model output. Numerical examples are presented, code is made available on GitHub, and discussions are provided to generate some useful insights of the proposed inverse modeling approaches.