$\phi-$DeepONet: A Discontinuity Capturing Neural Operator

1. Problem Statement

The paper addresses a critical limitation in current Scientific Machine Learning (SciML) and Neural Operator frameworks: the inability to accurately learn mappings between function spaces that contain discontinuities or exhibit non-smooth behavior.

• Context: Many physical phenomena involve interface problems where multiple materials, phases, or regimes interact (e.g., heat conduction in composites, flow in porous media). These problems are governed by partial differential equations (PDEs) with discontinuous coefficients and jump conditions at interfaces.
• Limitation of Existing Methods:
  • Standard Neural Operators (e.g., DeepONet): Based on the universal approximation theorem for continuous operators, they assume smooth input and output functions. They struggle to represent sharp gradients or discontinuities, leading to significant errors near interfaces.
  • Domain Decomposition Methods (e.g., IONet, XPINNs): While effective, these methods often require explicit domain decomposition, separate networks for each subdomain, or retraining when parameters change. This leads to high computational costs and scalability issues as the number of interfaces increases.
  • Physics-Informed Neural Networks (PINNs): Traditional PINNs solve for a single instance of a PDE and must be retrained for every change in material properties or boundary conditions, lacking the "operator" capability to generalize across parameter spaces.

2. Methodology: ϕ-DeepONet

The authors propose ϕ-DeepONet, a physics-informed neural operator designed to capture discontinuities without explicit domain decomposition. The framework modifies the standard DeepONet architecture (Branch and Trunk networks) to embed interface information directly into the solution representation.

Core Architecture Modifications

1. Multiple Branch Networks (Handling Discontinuous Inputs):

   • Instead of a single branch network, ϕ-DeepONet employs $P$ separate branch networks ($br_1, \dots, br_P$), where $P$ is the number of subdomains.
   • Each branch network processes the input function $u_q$ restricted to its specific subdomain $\Omega_q$. This allows the model to handle input fields that are themselves discontinuous across interfaces.

2. Augmented Trunk Network with Latent Embedding (Handling Discontinuous Outputs):

   • The trunk network, which typically outputs basis functions, is modified to accept an additional latent variable $\phi$.
   • Latent Embedding Construction: The spatial coordinates $y$ are mapped to a latent representation $\phi(y)$ that encodes the subdomain information. The paper explores three strategies:
     • Scalar Embedding (SE): Assigns a constant scalar value (e.g., the subdomain index) to each region.
     • Categorical Embedding (CE): Uses a one-hot encoding of the subdomain, projected into a low-dimensional latent space via a learnable embedding matrix $E$.
     • Non-linear Categorical Embedding (Non-linear CE): Applies a non-linear activation function (e.g., $\tanh$) to the projected one-hot vector, allowing the model to learn complex, non-linear relationships between the interface structure and the solution space.
   • The trunk network computes basis functions $tr_i(y, \phi(y))$ conditioned on both spatial location and the latent interface representation.

3. Solution Approximation:
   The final solution $\tilde{s}(y)$ is approximated as a linear combination of the branch outputs and the augmented trunk basis functions:
   $$\tilde{s}(y) \approx \sum_{i=1}^{K} \left( \prod_{q=1}^{P} br_i^q(u_q) \right) \cdot tr_i(y, \phi(y))$$
   This formulation allows the basis functions to "switch" behavior based on the latent variable $\phi$, effectively capturing discontinuities within a single continuous function space.

4. Training Strategy:

   • The model is trained using a physics- and interface-informed loss function ($L = L_{PDE} + L_{BC} + L_{I}$).
   • The loss enforces the PDE residual, boundary conditions, and crucially, the interface jump conditions (continuity of state and flux) via soft constraints (penalty terms).
   • Hard constraints for external boundaries can also be enforced via reparameterization.

3. Key Contributions

• Novel Architecture: Introduction of ϕ-DeepONet, which integrates interface information via a learnable latent embedding directly into the trunk network, eliminating the need for explicit domain decomposition or multiple trunk networks.
• Handling Discontinuous Inputs: Extension of the DeepONet framework to handle piecewise continuous input functions via multiple branch networks.
• Scalability: Demonstrates that the computational cost scales more favorably than domain-decomposition-based operators (like IONet) as the number of interfaces increases, avoiding the exponential growth in learnable parameters associated with separate networks per subdomain.
• Generalization: The method learns the operator mapping, meaning it can predict solutions for new instances of the PDE (different forcing functions or parameters) without retraining.

4. Results and Performance

The authors evaluated ϕ-DeepONet on 1D and 2D benchmark problems involving Poisson-type equations with discontinuous coefficients and inputs.

• Accuracy:

  • vs. Standard DeepONet: ϕ-DeepONet outperforms standard DeepONet by two orders of magnitude in relative $L_2$ error. Standard DeepONet fails to capture the sharp transitions at interfaces, while ϕ-DeepONet accurately resolves them.
  • vs. IONet: ϕ-DeepONet achieves comparable or superior accuracy to IONet (Interface Operator Network).
  • Embedding Types: Non-linear Categorical Embeddings generally provided the best accuracy, particularly in complex 2D scenarios, though Scalar Embeddings performed well in simpler 1D cases.

• Computational Efficiency:

  • ϕ-DeepONet is significantly more efficient than IONet. In the 1D multi-interface case, IONet required ~2.7x the training time of ϕ-DeepONet. In 2D cases, IONet was ~2.5x to 3.3x more expensive.
  • The method avoids the overhead of managing separate networks for each subdomain.

• Out-of-Distribution (OOD) Generalization:

  • The model demonstrated robustness when tested on input functions with different statistical properties (mean and length scales) than the training data.
  • Performance degraded slightly as the input length scale decreased (making the problem harder), but remained stable across variations in the mean.

• Complex Geometries:

  • The method successfully handled a 2D problem with a complex, petal-shaped interface, demonstrating its ability to generalize to non-trivial geometries without mesh generation.

5. Significance

• Bridging the Gap: ϕ-DeepONet bridges the gap between the flexibility of neural operators and the rigorous requirements of interface problems, which are ubiquitous in engineering and physics.
• Mesh-Free Advantage: By avoiding explicit domain decomposition and mesh generation, the method retains the mesh-free advantage of PINNs, simplifying preprocessing for complex geometries.
• Efficiency: It offers a computationally efficient alternative to existing domain-decomposition neural operators, making it viable for problems with many interfaces or complex topologies.
• Future Impact: This approach paves the way for learning operators in multiphysics systems with evolving interfaces, phase transitions, and heterogeneous materials, where traditional numerical methods struggle with mesh adaptation and where standard neural operators fail to converge.

Limitations & Future Work:
The current method relies on prior knowledge of subdomain partitions to construct the categorical embedding. Future work aims to develop data-driven methods to learn the interface structure implicitly without explicit partitioning, as well as extending the framework to stochastic interface problems.