arXiv:2512.01370v1 Announce Type: cross 
Abstract: Diffusion-based solvers for partial differential equations (PDEs) are often bottle-necked by slow gradient-based test-time optimization routines that use PDE residuals for loss guidance. They additionally suffer from optimization instabilities and are unable to dynamically adapt their inference scheme in the presence of noisy PDE residuals. To address these limitations, we introduce PRISMA (PDE Residual Informed Spectral Modulation with Attention), a conditional diffusion neural operator that embeds PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain, enabling gradient-descent free inference. In contrast to previous methods that use PDE loss solely as external optimization targets, PRISMA integrates PDE residuals as integral architectural features, making it inherently fast, robust, accurate, and free from sensitive hyperparameter tuning. We show that PRISMA has competitive accuracy, at substantially lower inference costs, compared to previous methods across five benchmark PDEs, especially with noisy observations, while using 10x to 100x fewer denoising steps, leading to 15x to 250x faster inference.

تم تقديم طريقة جديدة تُدعى PRISMA (تعديل طيفي مستنير بمخلفات PDE مع انتباه) لتحسين الحلول القائمة على الانتشار لمعادلات التفاضل الجزئي (PDE). تتضمن هذه الطريقة دمج مخلفات PDE مباشرة في هيكل النموذج باستخدام آليات الانتباه، مما يسمح بالاستدلال بدون انحدار تدريجي ومعالجة مشكلات عدم الاستقرار في التحسين وبطء روتين التحسين في وقت الاختبار.

Se ha introducido un nuevo método llamado PRISMA (Modulación Espectral Informada por Residuos PDE con Atención) para mejorar los solucionadores basados en difusión para ecuaciones en derivadas parciales (EDP). Este enfoque integra los residuos PDE directamente en la arquitectura del modelo mediante mecanismos de atención, permitiendo una inferencia sin descenso de gradiente y abordando problemas de inestabilidad en la optimización y lentitud en las rutinas de optimización en tiempo de prueba.

Une nouvelle méthode appelée PRISMA (Modulation Spectrale Informée par les Résidus PDE avec Attention) a été introduite pour améliorer les solveurs basés sur la diffusion pour les équations aux dérivées partielles (EDP). Cette approche intègre directement les résidus PDE dans l'architecture du modèle en utilisant des mécanismes d'attention, permettant une inférence sans descente de gradient et abordant les problèmes d'instabilité d'optimisation et de lenteur des routines d'optimisation en temps de test.

A new method called PRISMA (PDE Residual Informed Spectral Modulation with Attention) has been introduced to enhance diffusion-based solvers for partial differential equations (PDEs). This approach integrates PDE residuals directly into the model's architecture using attention mechanisms, allowing for gradient-descent free inference and addressing issues of optimization instability and slow test-time optimization routines.

Beyond Loss Guidance: Using PDE Residuals as Spectral Attention in Diffusion Neural Operators

Was this article worth reading? Share it

Prdkit

One More Thing in AI

MindPrism AI