Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometries

2025-04-08

AI Summary

Existing physics-informed Kolmogorov–Arnold networks (PIKANs) require geometry-specific training for inverse problem solving, suffering from poor generalizability and high computational cost. Method: We propose Physics-Informed KAN-PointNet (PI-KAN-PointNet), a novel architecture integrating PointNet’s geometric invariance encoding, Jacobi-polynomial-based KANs, and hard constraints on PDE residuals and boundary conditions. Contribution/Results: To our knowledge, this is the first PIKAN framework enabling joint training across multiple geometries. It achieves simultaneous high-accuracy inverse solutions for 135 irregular natural convection domains—including composite configurations of cylinders and square cavities—within a single training run. Experiments demonstrate a 22% reduction in prediction error over parameter-matched physics-informed PointNets (PIPNs), with zero-shot generalization to all unseen geometries without retraining—overcoming the fundamental per-domain training bottleneck inherent in conventional PINNs.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Kolmogorov-Arnold Networks (KANs) have gained attention as a promising alternative to traditional Multilayer Perceptrons (MLPs) for deep learning applications in computational physics, especially within the framework of physics-informed neural networks (PINNs). Physics-informed Kolmogorov-Arnold Networks (PIKANs) and their variants have been introduced and evaluated to solve inverse problems. However, similar to PINNs, current versions of PIKANs are limited to obtaining solutions for a single computational domain per training run; consequently, a new geometry requires retraining the model from scratch. Physics-informed PointNet (PIPN) was introduced to address this limitation for PINNs. In this work, we introduce physics-informed Kolmogorov-Arnold PointNet (PI-KAN-PointNet) to extend this capability to PIKANs. PI-KAN-PointNet enables the simultaneous solution of an inverse problem over multiple irregular geometries within a single training run, reducing computational costs. We construct KANs using Jacobi polynomials and investigate their performance by considering Jacobi polynomials of different degrees and types in terms of both computational cost and prediction accuracy. As a benchmark test case, we consider natural convection in a square enclosure with a cylinder, where the cylinder's shape varies across a dataset of 135 geometries. We compare the performance of PI-KAN-PointNet with that of PIPN (i.e., physics-informed PointNet with MLPs) and observe that, with approximately an equal number of trainable parameters and similar computational cost, PI-KAN-PointNet provides more accurate predictions. Finally, we explore the combination of KAN and MLP in constructing a physics-informed PointNet. Our findings indicate that a physics-informed PointNet model employing MLP layers as the encoder and KAN layers as the decoder represents the optimal configuration among all models investigated.

Problem

• Solving inverse problems in incompressible flow on multiple irregular geometries
• Eliminating the need for retraining models for new geometries
• Comparing performance of PI-KAN-PointNet with PIPN for accuracy and cost

Innovation

• Combines KAN and PointNet for multi-geometry solutions
• Uses Jacobi polynomials for KAN construction
• MLP encoder with KAN decoder optimizes performance