Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation (2404.12355v3)

Abstract: Foundation models, such as LLMs, 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.

• The paper introduces PROSE-PDE, a transformer-based model that concurrently predicts spatiotemporal states and discovers governing PDE equations.
• It employs a bi-modal pipeline integrating numerical and symbolic inputs to effectively address both forward and inverse PDE problems.
• The model demonstrates robust extrapolation by accurately generalizing to phenomena like shock and rarefaction waves while achieving high performance metrics.

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

The paper introduces the PROSE-PDE model, a novel multi-modal foundation model designed to address the challenges of multi-operator learning for partial differential equations (PDEs). This research is positioned within the context of scientific computing, where the foundation models, akin to those in natural language processing, have yet to be deeply explored. The primary objective of PROSE-PDE is to concurrently predict future states of spatiotemporal systems and discern the underlying governing equations from these systems.

Key Contributions

PROSE-PDE represents a significant advancement in the field for several reasons:

1. Multi-Operator and Multi-Modal Learning:
   • PROSE-PDE is characterized by its capability to process both numerical inputs and symbolic equation guesses, facilitating the resolution of complex PDE systems.
   • It is the first approach employing transformer-based architecture to address forward and inverse problems for various PDE classes, distinguishing itself with its ability to handle multiple operators.
2. Extrapolation of Physical Features:
   • A major highlight of the paper is the comprehensive extrapolation studies demonstrating the model's ability to predict PDE solutions beyond its training set. This includes the generalization to new physical phenomena and unseen parameter values, evidencing the model's robustness.
   • The paper specifically highlights the model's ability to extrapolate significant physical behavior such as shock and rarefaction waves in conservation laws, even when these phenomena are not explicitly part of the training set.
3. Strong Empirical Performance:
   • The model consistently achieves low prediction errors and high $R^2$ scores across various PDE types. These are robust indicators of the predictive competency of PROSE-PDE over its trained dataset and beyond.
   • The paper also reports rigorous ablation studies which reinforce the impact of its bi-modal architecture on performance stability and robustness against variations in training setups and data inputs.

Methodological Framework

PROSE-PDE's architecture is distinctly multi-modal, incorporating a dual pipeline for data and symbolic inputs. This design enables the coherent fusion of numerical simulations with symbolic information, which is pivotal in resolving well-posedness issues in multi-operator settings. The workflow proceeds by encoding both input modalities via distinct encoders, fusing them through a feature fusion block, and employing transformers for decoding the fused features into meaningful and accurate predictions.

Implications and Future Directions

The development of PROSE-PDE heralds a new avenue in scientific computing where models can transcend beyond discrete operator learning to form a foundation for generalized PDE solutions. This not only enhances our computational toolkit but also opens up prospects for AI-driven exploration of multi-scale and chaotic systems, traditionally limited by the scarcity of experimental data.

Future research could extend this work by exploring higher-dimensional PDEs and incorporating real-world noisy data to further validate the model’s applicability. Additionally, scaling the model, akin to LLMs, to cover broader classes of scientific phenomena remains a promising direction.

In conclusion, the introduction of PROSE-PDE provides a robust framework for advancing the integration of machine learning into scientific computing, laying the groundwork for adaptive, intelligent, and generalizable models capable of addressing the diverse challenges posed by complex dynamical systems.