Handling topology changes in learned representations for neural PDE methods

Develop learned representations and associated architectures for neural operator models (such as Fourier Neural Operator, DeepONet, and Graph Neural Operator) and physics-informed neural networks that explicitly accommodate topology changes in PDE solutions, including phenomena such as phase transitions, crack initiation and propagation, and bubble coalescence, thereby overcoming the prevailing fixed-topology assumption in current neural operator approaches.

Background

The paper highlights that many parametric PDE scenarios involve topology changes, such as phase transitions in materials, fracture initiation and propagation in solids, and bubble coalescence in multiphase flows. These situations violate the common assumption of fixed topology embedded in most current neural operator approaches, which typically operate on static domains or meshes.

Accurately representing and learning solution behavior across topology-changing events is critical for extending neural methods to high-impact applications (e.g., fracture mechanics, multiphase fluid dynamics, and phase-field models). The authors explicitly state that representing topology changes in learned representations remains an open problem, underscoring a gap between current operator learning capabilities and practical needs in complex parametric physics.

Addressing this problem would enable robust generalization across parameter spaces where the solution manifold undergoes topological transitions, improving reliability and applicability of neural operators and PINNs in real-world multi-physics and design tasks.