Systems governed by partial differential equations (PDEs) require computationally intensive numerical solvers to predict spatiotemporal field evolution. While machine learning (ML) surrogates offer faster solutions, autoregressive inference with ML models suffer from error accumulation over successive predictions, limiting their long-term accuracy. We propose a deep ensemble framework to address this challenge, where multiple ML surrogate models with random weight initializations are trained in parallel and aggregated during inference. This approach leverages the diversity of model predictions to mitigate error propagation while retaining the autoregressive strategies ability to capture the system's time dependent relations. We validate the framework on three PDE-driven dynamical systems - stress evolution in heterogeneous microstructures, Gray-Scott reaction-diffusion, and planetary-scale shallow water system - demonstrating consistent reduction in error accumulation over time compared to individual models. Critically, the method requires only a few time steps as input, enabling full trajectory predictions with inference times significantly faster than numerical solvers. Our results highlight the robustness of ensemble methods in diverse physical systems and their potential as efficient and accurate alternatives to traditional solvers. The codes for this work are available on GitHub (https://github.com/Graham-Brady-Research-Group/AutoregressiveEnsemble_SpatioTemporal_Evolution).

Relation to Prior Work: Yes, it mostly covers all of the literature. It might make sense to update the reference to include latest content. Some of the papers I mentioned and some of the latest content: 1. Agrawal, H., Desai, K., Wang, Y., Chen, X., Jain, R., Johnson, M., ... & Anderson, P. (2019). Are we pretraining it right? Towards vqa models that can read. Oscar: Object-semantics aligned pre-training for vision-language tasks.

Deep operator network (DeepONet) has shown great promise as a surrogate model for systems governed by partial differential equations (PDEs), learning mappings between infinite-dimensional function spaces with high accuracy. However, achieving low generalization errors often requires highly overparameterized networks, posing significant challenges for large-scale, complex systems. To address these challenges, latent DeepONet was proposed, introducing a two-step approach: first, a reduced-order model is used to learn a low-dimensional latent space, followed by operator learning on this latent space. While effective, this method is inherently data-driven, relying on large datasets and making it difficult to incorporate governing physics into the framework. Additionally, the decoupled nature of these steps prevents end-to-end optimization and the ability to handle data scarcity. This work introduces PI-Latent-NO, a physics-informed latent operator learning framework that overcomes these limitations. Our architecture employs two coupled DeepONets in an end-to-end training scheme: the first, termed Latent-DeepONet, identifies and learns the low-dimensional latent space, while the second, Reconstruction-DeepONet, maps the latent representations back to the original physical space. By integrating governing physics directly into the training process, our approach requires significantly fewer data samples while achieving high accuracy. Furthermore, the framework is computationally and memory efficient, exhibiting nearly constant scaling behavior on a single GPU and demonstrating the potential for further efficiency gains with distributed training. We validate the proposed method on high-dimensional parametric PDEs, demonstrating its effectiveness as a proof of concept and its potential scalability for large-scale systems.

Big data is transforming scientific progress by enabling the discovery of novel models, enhancing existing frameworks, and facilitating precise uncertainty quantification, while advancements in scientific machine learning complement this by providing powerful tools to solve inverse problems to identify the complex systems where traditional methods falter due to sparse or noisy data. We introduce two innovative neural operator frameworks tailored for discovering hidden physics and identifying unknown system parameters from sparse measurements. The first framework integrates a popular neural operator, DeepONet, and a physics-informed neural network to capture the relationship between sparse data and the underlying physics, enabling the accurate discovery of a family of governing equations. The second framework focuses on system parameter identification, leveraging a DeepONet pre-trained on sparse sensor measurements to initialize a physics-constrained inverse model. Both frameworks excel in handling limited data and preserving physical consistency. Benchmarking on the Burgers' equation and reaction-diffusion system demonstrates state-of-the-art performance, achieving average $L_2$ errors of $\mathcal{O}(10^{-2})$ for hidden physics discovery and absolute errors of $\mathcal{O}(10^{-3})$ for parameter identification. These results underscore the frameworks' robustness, efficiency, and potential for solving complex scientific problems with minimal observational data.

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate mappings between infinite-dimensional Banach spaces. As data-driven models, neural operators require the generation of labeled observations, which in cases of complex high-fidelity models result in high-dimensional datasets containing redundant and noisy features, which can hinder gradient-based optimization. Mapping these high-dimensional datasets to a low-dimensional latent space of salient features can make it easier to work with the data and also enhance learning. In this work, we investigate the latent deep operator network (L-DeepONet), an extension of standard DeepONet, which leverages latent representations of high-dimensional PDE input and output functions identified with suitable autoencoders. We illustrate that L-DeepONet outperforms the standard approach in terms of both accuracy and computational efficiency across diverse time-dependent PDEs, e.g., modeling the growth of fracture in brittle materials, convective fluid flows, and large-scale atmospheric flows exhibiting multiscale dynamical features.

Tilottama Goswami has received a BE degree with Honors in Computer Science and Engineering from the National Institute of Technology, Durgapur; and an MS degree in Computer Science (High Distinction) from Rivier University, Nashua, New Hampshire, United States. She was awarded a PhD in Computer Science from the University of Hyderabad. Presently, Dr. Goswami is Professor in the Department of Information Technology, Vasavi College of Engineering, Hyderabad, India. She has, overall, 23 years of experience in academia, research, and the IT industry. Her research interests are computer vision, machine learning, and image processing.