Time-Series Forecasting, Knowledge Distillation, and Refinement within a Multimodal PDE Foundation Model

Symbolic encoding has been used in multi-operator learning as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations, the equation itself provides an additional modality to identify the system. The utilization of symbolic expressions along side time-series samples allows for the development of multimodal predictive neural networks. A key challenge with current approaches is that the symbolic information, i.e. the equations, must be manually preprocessed (simplified, rearranged, etc.) to match and relate to the existing token library, which increases costs and reduces flexibility, especially when dealing with new differential equations. We propose a new token library based on SymPy to encode differential equations as an additional modality for time-series models. The proposed approach incurs minimal cost, is automated, and maintains high prediction accuracy for forecasting tasks. Additionally, we include a Bayesian filtering module that connects the different modalities to refine the learned equation. This improves the accuracy of the learned symbolic representation and the predicted time-series.

PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

LeMON: Learning to Learn Multi-Operator Networks

Single-operator learning involves training a deep neural network to learn a specific operator, whereas recent work in multi-operator learning uses an operator embedding structure to train a single neural network on data from multiple operators. Thus, multi-operator learning is capable of predicting a range of operators within one model. In this work, we propose pretraining and fine-tuning strategies for solving PDEs using multi-operator learning. One key aspect is that by increasing the number of families of operators used in pretraining, a PDE foundation model can be fine-tuned to downstream tasks involving new PDEs with a limited number of samples, thus outperforming single operator neural networks. Specifically, a multi-operator learning model pre-trained with data from diverse PDE families can predict unseen operators after fine-tuning with only a limited number of operators from the new family, enabling them to serve as a data-free PDE solver. We also show that the proposed training and fine-tuning method is able to predict new operators in zero-shot prediction without samples. Additionally, we introduce a PDE-agnostic meta-learning algorithm to improve the adaptability of the model to various PDEs by providing a better parameter initialization process. To address the needs of applications with limited computing resources, we explore low-rank adaptation methods that reduce computational costs while enhancing solver accuracy. Lastly, by examining the scaling law with respect to the number of operator families, we establish and highlight its potential for broad adaptation in PDE-solving tasks.

Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation

Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a multi-operator learning approach which can predict future states of spatiotemporal systems while concurrently learning the underlying governing equations of the physical system. Specifically, we focus on multi-operator learning by training distinct one-dimensional time-dependent nonlinear constant coefficient partial differential equations, with potential applications to many physical applications including physics, geology, and biology. More importantly, we provide three extrapolation studies to demonstrate that PROSE-PDE can generalize physical features through the robust training of multiple operators and that the proposed model can extrapolate to predict PDE solutions whose models or data were unseen during the training. Furthermore, we show through systematic numerical experiments that the utilization of the symbolic modality in our model effectively resolves the well-posedness problems with training multiple operators and thus enhances our model's predictive capabilities.