Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators (2404.10843v1)

Abstract: We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

• The paper introduces a novel GNP framework that learns geometric properties from manifold data using layered operator mechanisms.
• The paper demonstrates effective approximation of Laplace-Beltrami operators and reliable solution maps for PDEs on complex geometries.
• The paper showcases applications in Bayesian inverse problems, enabling precise shape estimation on irregular, non-flat domains.

Advancements in Geometric Neural Operators for Learning from Manifold Data

Introduction to Geometric Neural Operators (GNPs)

Geometric Neural Operators (GNPs) introduce a groundbreaking approach to learning operations that account for geometric contributions in data-driven problems. Unlike traditional models that operate primarily within Euclidean domains, GNPs handle arbitrary-shaped domains, curved surfaces, and other manifold representations efficiently. These operators extend the scope of neural operator frameworks by focusing on the rich geometric structures inherent in many scientific and engineering problems.

Core Contributions and Methodological Setup

GNPs are designed to:

• Estimate geometric properties such as metrics and curvatures directly from data, particularly from point-cloud representations.
• Approximate and learn solution maps for differential operators, specifically Laplace-Beltrami (LB) operators on manifold surfaces.
• Solve complex Bayesian inverse problems for identifying manifold shapes based on observational data.

In terms of architecture, GNPs leverage a combination of lifting procedures, integral and local linear operators, and non-linear activation functions to process input functions and geometric information. The model flexibly alternates between capturing local and global geometric features by stacking multiple operator layers, which incrementally expand the receptive field across the architecture.

Experiments and Results

The paper extensively evaluates GNPs on several tasks:

1. Learning Geometric Quantities: GNPs succeed in learning first and second fundamental forms of geometry from point-cloud data. This is critical for tasks where understanding the inherent geometry of data is pivotal, such as in shape analysis and physical simulations on complex surfaces.
2. Solving PDEs on Manifolds: The GNPs are adept at learning solution maps for Laplace-Beltrami-Poisson problems across different manifold geometries. The ability to handle various manifold topologies and shapes underscores the flexibility and robustness of the proposed approach.
3. Bayesian Inverse Problems: GNPs effectively solve inverse problems to identify manifold shapes based on observations derived from differential operators. Through a combination of geometric deep learning and Bayesian inference, the models predict manifold shapes with high accuracy, demonstrating the potential of GNPs in fields like medical imaging and computer vision where shape estimation is crucial.

Practical Implications and Theoretical Significance

The development of GNPs marks a significant step towards bridging the gap between geometric data representations and machine learning models. Practically, GNPs can impact various applications, including dynamic system modeling on irregular domains, complex material behavior analysis on non-flat surfaces, and advanced image processing tasks involving intricate topological structures.

Theoretically, the integration of geometry into learning operators presents a novel paradigm in the extrapolation capabilities of neural networks. It offers a refined understanding of how deep learning can adapt to manifold-valued data, paving the way for more generalized and theoretically sound models.

Future Prospects and Enhancements

Future work could explore several enhancements to GNPs:

• Scalability and Efficiency: Optimizing the computational efficiency for high-dimensional manifolds and large datasets remains a challenge.
• Integration with Other Learning Paradigms: Combining GNPs with reinforcement learning or unsupervised learning could provide deeper insights into many physical and biological processes.
• Theoretical Bounds and Guarantees: Establishing theoretical performance bounds for GNPs could strengthen their reliability for critical applications in science and engineering.

Conclusion

GNPs represent a transformative approach to incorporating geometric insights into machine learning models. By effectively handling the geometric intricacies of data and operations on manifolds, GNPs open new avenues in modeling, analysis, and inference across diverse fields.