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Geometry-Informed Neural Networks (2402.14009v3)

Published 21 Feb 2024 in cs.LG and cs.CV

Abstract: Geometry is a ubiquitous tool in computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative learning strategies. To this end, we introduce geometry-informed neural networks (GINNs) -- a framework for training shape-generative neural fields without data by leveraging user-specified design requirements in the form of objectives and constraints. By adding diversity as an explicit constraint, GINNs avoid mode-collapse and can generate multiple diverse solutions, often required in geometry tasks. Experimentally, we apply GINNs to several validation problems and a realistic 3D engineering design problem, showing control over geometrical and topological properties, such as surface smoothness or the number of holes. These results demonstrate the potential of training shape-generative models without data, paving the way for new generative design approaches without large datasets.

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Summary

  • The paper introduces GINNs, a novel approach leveraging geometric constraints for data-free generative modeling.
  • It employs neural fields and a connectedness loss to ensure diverse, structurally sound outputs across various design tasks.
  • Experimental validations in 2D and 3D scenarios demonstrate its robustness, advancing constrained optimization in generative design.

Geometry-Informed Neural Networks

This paper introduces the concept of Geometry-Informed Neural Networks (GINNs), a novel method that leverages geometric constraints to guide the learning process of neural networks in generative modeling tasks. GINNs uniquely operate without necessitating training data, instead relying on predefined geometric constraints to generate solutions. The authors propose an explicit diversity loss to curb mode collapse, a common issue in generative modeling where the generator produces less varied outputs than desired. This innovative approach is evaluated experimentally across a variety of two and three-dimensional scenarios, demonstrating its efficacy and robustness.

Key Contributions

  1. Introduction of GINNs: The paper presents an overarching framework for GINNs, which are designed to produce solutions constrained by geometric properties without relying on empirical data. This framework is particularly suited for applications where data is scarce but geometric constraints and objectives are well-defined.
  2. Neural Fields as Representation: The authors advocate for the use of neural fields, particularly implicit neural shapes (INSs), as the primary representation for geometric objects. Neural fields are compact, continuous, and differentiable representations that offer advantages over traditional discrete methods.
  3. Handling Under-Determined Systems: Recognizing that many geometric problems are under-determined and admit multiple solutions, the paper asserts the necessity for generating diverse sets of solutions. This is facilitated by incorporating a diversity loss to the training process.
  4. Connectedness Constraint: A significant portion of the work is dedicated to formulating a connectedness constraint, which ensures that generated shapes form a single connected component. This is achieved through the application of Morse theory, translating the connectedness property into a differentiable loss.
  5. Experimental Validation: The efficacy of GINNs is validated through experiments on classic problems such as Plateau's problem, as well as realistic 3D engineering problems like the design of jet engine brackets. These experiments highlight the method's ability to generate diverse, high-quality solutions under complex geometric constraints.

Numerical Results and Claims

Several numerical evaluations demonstrate the power and versatility of GINNs:

  • Plateau's Problem: The GINNs successfully produced minimal surfaces constrained by a given boundary, showcasing their capability to handle well-posed geometric problems.
  • Parabolic Mirror Design: For an optical design problem, GINNs were able to identify a surface that directs reflected rays to a focal point, approximating the ideal parabolic shape.
  • Obstacle Course Problem: A GINN trained to connect two interfaces around an obstacle demonstrated the necessity and effectiveness of the connectedness loss. The generative GINN produced multiple viable connection paths, emphasizing the importance of diversity in such tasks.
  • Jet Engine Bracket: In a 3D scenario inspired by a real-world engineering challenge, GINNs generated diverse, viable designs for a jet engine bracket, adhering to a complex set of geometric constraints while maintaining structural integrity.

Theoretical and Practical Implications

The theoretical implications of this research extend into several domains:

  • Generative Design: GINNs present a significant advancement in the field of generative design, particularly in areas where traditional methods are infeasible due to data scarcity.
  • Optimization in High Dimensions: The approach demonstrates potential improvements in optimizing high-dimensional spaces where constraints can be efficiently embedded into the learning process.
  • Constraint-Driven Neural Networks: The integration of geometric constraints into neural network training opens avenues for other types of constraints, such as physical laws and topological properties.

On a practical note, the implementation of GINNs in design processes can streamline the development of innovative solutions, providing engineers and designers with a powerful tool to explore vast solution spaces efficiently.

Future Directions

The results invite several future research directions:

  • Extension to Other Constraints: Expanding the framework to incorporate a broader range of constraints, including those from related domains like physics and topology, could enhance the versatility of GINNs.
  • Improving Computational Efficiency: Optimizing the computation of complex losses, particularly the connectedness loss, through more efficient algorithms or approximations, could significantly reduce training times.
  • Advanced Conditioning Mechanisms: Investigating alternative methods for latent space conditioning in neural fields might improve the structure and quality of generated solutions.
  • Scalability and Robustness: Scaling up the experiments to higher-dimensional and more complex scenarios while ensuring solution robustness will be crucial for real-world applicability.

In summary, this paper establishes the foundation for geometry-informed neural networks, pushing the boundaries of generative modeling driven by constraints rather than data. This represents a promising direction for future research in machine learning and its applications in engineering and design.

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