Technical Summary: Level-Set Physics-Informed Neural Networks for Domain Inverse Problems of Gravimetry

1. Problem Statement

The paper addresses the domain inverse problem of gravimetry, which aims to reconstruct the geometry of subsurface anomalous mass distributions (domain $D$) from surface measurements of gravity acceleration. While classical gravimetry is notoriously ill-posed (non-unique), the authors focus on a specific "correctness class" where the density contrast $f$ is known a priori, and the unknown is the characteristic function $\chi_D$ of the domain.

The primary challenge in applying standard Physics-Informed Neural Networks (PINNs) to this problem is the discontinuous nature of the interface between the anomalous domain and the background. Directly representing the density $\rho = f \chi_D$ via a neural network is problematic because:

1. Neural networks inherently produce continuous outputs, making it difficult to represent sharp, discontinuous interfaces.
2. The evolution of these sharp interfaces during training leads to exploding or vanishing gradients, particularly when using numerical approximations of the Heaviside function to model the discontinuity.

2. Methodology

The authors propose Level-Set Physics-Informed Neural Networks (Level-Set PINNs) to overcome these limitations. The methodology consists of the following core components:

2.1 Level-Set Formulation

Instead of directly predicting the density, the method uses a neural network $\phi_\omega(x)$ to represent a continuous level-set function. The unknown domain $D$ is defined implicitly by the zero level-set of this function:
$$\rho_\omega(x) = f(x) H_\tau(\phi_\omega(x))$$
where $H_\tau$ is a numerical (smoothed) Heaviside function controlled by a parameter $\tau$. Ideally, $\phi_\omega$ approximates a signed-distance function, where $\{x \mid \phi_\omega(x) = 0\}$ represents the sharp interface $\partial D$.

2.2 Loss Function Construction

The training minimizes a composite loss function $L_{total}$ that enforces physical laws and data fidelity:

• PDE Residual ($L_{pde}$): Enforces the Poisson equation $-\Delta U = \gamma \rho$.
• Boundary Data ($L_{bc}$): Fits the measured gravity acceleration $g = \nabla U$ on the observation surface.
• Far-Field Condition ($L_{far}$): Enforces the decay of the potential at infinity.
• Level-Set Regularization:
  • Eikonal Term ($L_{eik}$): Penalizes deviations from $|\nabla \phi| = 1$ to maintain the signed-distance property.
  • Length/Area Penalty ($L_r$): Penalizes the gradient norm to prevent structural irregularities (e.g., burrs) and promote stable evolution.
• Parameter Regularization ($L_\Theta$): Standard weight decay.

2.3 Interface-Aware Backpropagation

A critical innovation is the interface-aware backpropagation strategy to handle the gradient issues caused by the sharp interface.

• The Dilemma: A small $\tau$ is needed for a sharp interface (accuracy), but it causes exploding gradients at the zero level-set and vanishing gradients elsewhere (training instability).
• The Solution: The authors decouple the forward and backward passes.
  • Forward Pass: Uses a small $\tau$ to maintain a sharp density interface.
  • Backward Pass: Redefines the derivative of the Heaviside function using a larger parameter $\tilde{\tau} = c\tau$ (where $c > 1$).
• Theoretical Justification: The paper provides a rigorous analysis (Proposition 3.1) proving that minimizing the modified loss term with $\tilde{\tau}$ closely approximates minimizing the true loss with $\tau$, provided $\tau$ is small. This creates a "broader support region" for gradient flow without sacrificing the sharpness of the final interface.

2.4 Adaptive Refinement and Architecture

• Adaptive Refinement: An iterative strategy dynamically adds collocation points near the evolving interface (where $|\phi_\omega(x)| < C_{refine}$) to improve resolution where it is most needed.
• Network Architecture:
  • Level-Set Network ($\phi_\omega$): Uses ReLU activation. ReLU is chosen for its non-saturating nature and efficiency in approximating $W^{1,\infty}$ functions (signed-distance functions).
  • Potential Network ($U_\theta$): Uses tanh activation. Since the PDE loss requires the Laplacian $\Delta U$, a smooth activation is necessary because the second derivative of ReLU vanishes almost everywhere.

2.5 Initialization Strategy

To mitigate sensitivity to initialization in ill-posed problems, a two-step pre-training is employed:

1. Pre-train $\phi_\omega$ to match a geometric initial guess.
2. Pre-train $U_\theta$ to satisfy the PDE given the initialized density.

3. Key Contributions

1. Level-Set PINN Framework: The first application of level-set methods within the PINN paradigm specifically for domain inverse problems in gravimetry, avoiding the need to lift dimensions (unlike previous forward-only interface methods).
2. Interface-Aware Backpropagation: A novel training strategy that resolves the exploding/vanishing gradient dilemma in level-set PINNs by redefining derivatives during backpropagation, supported by detailed mathematical error analysis.
3. Adaptive Collocation: A mesh-free strategy to refine sampling points near the interface dynamically during training.
4. Theoretical Analysis: Proofs regarding the approximation capabilities of ReLU and tanh networks for the specific Sobolev spaces ($W^{1,\infty}$) relevant to level-set functions and gravitational potentials.

4. Numerical Results

The method is validated through extensive 2D and 3D numerical examples:

• 2D Example (Circle and Parallelogram): The method successfully recovers sharp boundaries of complex shapes. Baseline comparisons show that standard PINNs (direct density representation) fail to recover meaningful geometry, and even PINNs with sigmoid constraints perform significantly worse than the proposed Level-Set PINNs.
• 3D Examples:
  • Four Spherical Anomalies: Successfully recovers four distinct spherical bodies with sharp interfaces.
  • Mixed Geometries: Recovers a dipping parallelepiped and other complex shapes.
  • Depth-Dependent Density: Successfully handles cases where the density contrast $f(x)$ varies with depth, not just the geometry.
• Performance: In all cases, the predicted gravity data matches the measured data with high fidelity, and the reconstructed domains exhibit sharp, well-defined interfaces consistent with the true models.

5. Significance and Claims

The paper claims that Level-Set PINNs provide a robust and effective solution for the domain inverse problem of gravimetry, a class of problems where traditional PINNs struggle due to discontinuities.

• Correctness Class: The work reinforces the utility of restricting inverse gravimetry to the "correctness class" (domain inverse problems) where uniqueness is theoretically guaranteed, rather than attempting to solve the fully ill-posed general inverse problem.
• Generalizability: While focused on gravimetry, the authors suggest the methodology offers broader insights for general domain and interface reconstruction problems involving discontinuous structures.
• Future Direction: The paper modestly notes that future work could explore embedding physical laws directly into unrolled neural network layers (model-guided architectures), but does not propose specific new applications beyond the scope of the current study.

The authors emphasize that their approach does not rely on massive training datasets (unlike end-to-end deep learning) but rather leverages physical laws and geometric constraints to ensure reliability and generalization.