Adaptive Mesh-Quantization for Neural PDE Solvers

2025-11-23

AI Summary

Uniform computational resource allocation across all mesh nodes in graph neural PDE solvers leads to inefficiency, as it ignores spatially heterogeneous physical complexity. Method: We propose a spatially adaptive quantization framework that employs a lightweight auxiliary model to dynamically identify high-error regions and accordingly assign differentiated bitwidths to node, edge, and cluster features. The method integrates state-of-the-art graph neural architectures—including MP-PDE and GraphViT—to enable PDE solving on unstructured meshes. Contribution/Results: Our approach achieves Pareto-optimal accuracy–cost trade-offs across diverse physical field tasks. Under identical computational budgets, it improves solution accuracy by up to 50% over uniform quantization baselines, while significantly enhancing resource utilization efficiency and generalization capability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Physical systems commonly exhibit spatially varying complexity, presenting a significant challenge for neural PDE solvers. While Graph Neural Networks can handle the irregular meshes required for complex geometries and boundary conditions, they still apply uniform computational effort across all nodes regardless of the underlying physics complexity. This leads to inefficient resource allocation where computationally simple regions receive the same treatment as complex phenomena. We address this challenge by introducing Adaptive Mesh Quantization: spatially adaptive quantization across mesh node, edge, and cluster features, dynamically adjusting the bit-width used by a quantized model. We propose an adaptive bit-width allocation strategy driven by a lightweight auxiliary model that identifies high-loss regions in the input mesh. This enables dynamic resource distribution in the main model, where regions of higher difficulty are allocated increased bit-width, optimizing computational resource utilization. We demonstrate our framework's effectiveness by integrating it with two state-of-the-art models, MP-PDE and GraphViT, to evaluate performance across multiple tasks: 2D Darcy flow, large-scale unsteady fluid dynamics in 2D, steady-state Navier-Stokes simulations in 3D, and a 2D hyper-elasticity problem. Our framework demonstrates consistent Pareto improvements over uniformly quantized baselines, yielding up to 50% improvements in performance at the same cost.

Problem

• Addresses inefficient uniform computation in neural PDE solvers
• Proposes adaptive bit-width allocation for varying spatial complexity
• Optimizes resource use by focusing computation on high-loss regions

Innovation

• Adaptive bit-width allocation for mesh quantization
• Lightweight auxiliary model identifies high-loss regions
• Dynamic resource distribution optimizes computational utilization