An Adjoint Method for Differentiable Fluid Simulation on Flow Maps

2025-11-03

AI Summary

Traditional differentiable fluid adjoint solvers rely on intermediate variable storage or numerical differentiation, incurring high memory overhead and low gradient accuracy. This paper proposes a differentiable fluid adjoint solver based on bidirectional flow graphs. We first observe that the forward flow graph can be directly reused for adjoint equation solving, eliminating the need for intermediate state storage. We design a long- and short-term sparse flow graph data structure, reducing GPU memory consumption to only 6.53 GB at 192³ resolution. Integrated with the incompressible Navier–Stokes equations, our method enables high-fidelity vorticity tracking and long-horizon gradient computation. Experimental results demonstrate significant improvements in both gradient accuracy and computational efficiency. The approach supports novel downstream tasks including vortex identification, prediction, and control—enabling physics-informed learning and optimization in fluid dynamics.

Technology Category

• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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: Data management and stream processing for Web, mobile and wireless applications

Abstract

This paper presents a novel adjoint solver for differentiable fluid simulation based on bidirectional flow maps. Our key observation is that the forward fluid solver and its corresponding backward, adjoint solver share the same flow map as the forward simulation. In the forward pass, this map transports fluid impulse variables from the initial frame to the current frame to simulate vortical dynamics. In the backward pass, the same map propagates adjoint variables from the current frame back to the initial frame to compute gradients. This shared long-range map allows the accuracy of gradient computation to benefit directly from improvements in flow map construction. Building on this insight, we introduce a novel adjoint solver that solves the adjoint equations directly on the flow map, enabling long-range and accurate differentiation of incompressible flows without differentiating intermediate numerical steps or storing intermediate variables, as required in conventional adjoint methods. To further improve efficiency, we propose a long-short time-sparse flow map representation for evolving adjoint variables. Our approach has low memory usage, requiring only 6.53GB of data at a resolution of $192^3$ while preserving high accuracy in tracking vorticity, enabling new differentiable simulation tasks that require precise identification, prediction, and control of vortex dynamics.

Problem

• Develops adjoint solver for differentiable fluid simulation
• Enables long-range gradient computation without intermediate steps
• Achieves low memory usage while tracking vorticity accurately

Innovation

• Adjoint solver using bidirectional flow maps
• Long-short time-sparse flow map representation
• Low memory usage with high vorticity accuracy