DiffFluid: Diffuse Interface & Neural Fluid Models

• DiffFluid is a comprehensive framework that employs diffuse interface methods to model multiphase and multi-material flows using both traditional PDE solvers and neural networks.
• It leverages advanced discretization techniques, adaptive mesh refinement, and differentiable programming to achieve high-fidelity simulations in challenging fluid dynamics and optimization tasks.
• Recent innovations include DDPM+Transformer architectures that offer state-of-the-art accuracy in predicting fluid flows, with significant improvements demonstrated on benchmark problems.

DiffFluid is a term that denotes a range of frameworks and methodologies based on the concept of “diffuse interface” treatments in computational fluid dynamics, encompassing both deep-learning-driven flow predictors and PDE-based multiphysics solvers. Recent lines of research under the DiffFluid label include: physics-based diffuse interface models (Navier–Stokes–Cahn–Hilliard, Baer–Nunziato, multiphase reactive-elastoplastic systems), fully differentiable rigid/multiphase simulators for robot learning, and neural surrogate models (notably DDPM + Transformer architectures) for solving high-dimensional PDEs in fluid mechanics. These technologies enable robust modeling of multi-phase, multi-material phenomena and gradient-based optimization in scientific, engineering, and robotics applications.

1. Diffuse Interface Theory and Classical DiffFluid Models

Diffuse interface (DI) models, also referred to as phase-field methods, resolve material or phase interfaces as continuous transitions of an order parameter, rather than sharp discontinuities. In the context of binary or multi-component fluids, the system's free energy is typically given by a Ginzburg–Landau functional:

$$F[\phi] = \int_\Omega \left( \frac{\sigma}{\epsilon} \psi(\phi) + \frac{\sigma\,\epsilon}{2} |\nabla\phi|^2 \right) dx,$$

where $\phi$ is the order parameter and $\epsilon$ controls interface thickness.

The governing equations for isothermal incompressible two-phase flow are the coupled Navier–Stokes–Cahn–Hilliard (NSCH) system:

• Cahn–Hilliard phase evolution:

$$\partial_t \phi + \mathbf{u}\cdot\nabla\phi = \nabla\cdot\left( m(\phi) \nabla\mu \right),$$

with $\mu = \delta F/\delta \phi$ the chemical potential.

• Navier–Stokes momentum balance with capillarity:

$$\rho(\phi)\left(\partial_t\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right) = -\nabla p + \nabla\cdot(2\eta(\phi) D(\mathbf{u})) + \mu\nabla\phi,$$

where $D(\mathbf{u})$ is the symmetrized gradient.

Diffuse interface methods provide a thermodynamically consistent, variational framework, with energy dissipation directly derived from their structure. As $\epsilon\to 0$, sharp-interface limits recover classical two-phase Navier–Stokes equations with physically-consistent free boundary conditions, including Gibbs–Thomson and stress-jump laws (Abels et al., 2010, Demont et al., 2022). DI techniques have been rigorously extended to compressible flows and multi-material reactive-elastoplastic systems (Kemm et al., 2020, Wallis et al., 2020).

2. High-Order Diffuse Interface Solvers

Modern DI-based simulation frameworks employ advanced discretizations and solver strategies. For binary-fluid flows, an established methodology is the AGG NSCH model featuring adaptive finite element mesh refinement with hierarchical THB-splines, $\varepsilon$-continuation interface regularization, and robust time/inexact-Newton solvers. These allow evolution of extremely fine, physically-consistent multiphase interfaces, validated against sharp-interface analytical and reference solutions (Demont et al., 2022).

For compressible or multi-phase flows around moving solids of arbitrary shape, the reduced Baer–Nunziato approach encodes solids/fluids via a scalar volume fraction $\alpha(x,t)$, sidestepping explicit tracking of interfaces. The governing system is advanced using path-conservative high-order ADER-DG methods with subcell finite-volume limiters for strict conservation and robust shock capturing. The normal velocity continuity at the interface emerges from the model's mathematical structure, confirmed by both Riemann invariant analysis and generalized Rankine–Hugoniot theory (Kemm et al., 2020).

In reactive multi-material systems, a unified conservative PDE system couples mass, momentum, energy, and state variables for all components, with interface steepening (e.g., THINC–BVD) and adaptive mesh refinement frameworks (AMReX base) ensuring stability, accuracy, and conservation even under large deformations and shock-dominated dynamics (Wallis et al., 2020).

3. Differentiable and Gradient-Based DiffFluid Frameworks

For simulation-driven optimization and learning, fully differentiable DI physics engines have emerged. FluidEngine—the core of FluidLab—integrates hybrid particle/grid-based MPMs with grid-based solvers, supporting solids, liquids (Newtonian, non-Newtonian), and gases. Written in Taichi with custom autograd, it enables backpropagation over thousands of steps with fine-grained gradient checkpointing, supporting reinforcement learning, end-to-end trajectory optimization, and multi-material coupling (Xian et al., 2023).

Several domain-specific schemes address the nonconvex and non-smooth nature of fluid optimization: soft contact blending for contact discontinuities, temporally expanding horizons for progressive long-horizon tasks, and fluid continuity-aware gradient sharing. These mechanisms, together with the differentiable backend, have demonstrated strong empirical performance on canonical and complex robotic manipulation tasks (e.g., latte art, ice-cream scooping) and sim-to-real transfer using a 7-DoF Franka robot arm.

4. Neural DiffFluid Models: DDPM-Based Fluid Prediction

Recent innovation under the DiffFluid moniker is the reformulation of fluid-dynamical PDE prediction as an image-to-image translation task using denoising diffusion probabilistic models (DDPMs) with Transformer-based backbones. In this approach, both input geometry/physical fields and target solution fields are discretized as high-dimensional multi-channel "images." The forward process adds Gaussian or multi-scale noise, and a neural model is trained to recover the noise component, enabling probabilistic sampling of fluid flow fields.

A canonical implementation uses stacked Diffusion Transformer (DiT) blocks, patch embeddings, and adaptive normalization. Enhancements such as multi-resolution noise and annealing significantly improve sharp interface/boundary capture. Loss functions mix mean-square error on noise with L1 error on the reconstructed solutions, promoting both global smoothness and sharp local features (Luo et al., 2024, Gachnang et al., 10 Jul 2025, Yang et al., 2023).

Notably, DiffFluid achieves state-of-the-art relative L2 errors (\textasciitilde5\% for Navier–Stokes, \textasciitilde0.5\% for Darcy, and airfoil benchmarks), robustly outperforming comparable neural operator architectures (Transolver, FNO) particularly in high Reynolds number and multi-scale regimes (Luo et al., 2024). Replication studies confirm the potential of diffusion-based neural fluid solvers and their generalizability to new PDEs, including Lattice Boltzmann (Gachnang et al., 10 Jul 2025).

Benchmark	DiffFluid Rel. L2 Error	Gain vs. Baseline
Navier–Stokes	0.0497	+44.8%
Darcy	0.0049	+14.0%
Airfoil (Euler)	0.0047	+11.3%

Multi-resolution noise, annealing, and hybrid loss are ablated to quantify effects on sharp feature reconstruction and accuracy. The methodology demonstrates flexibility, but training/inference costs are significant due to $O(T)$ model evaluations.

5. Multiphysics DiffFluid Frameworks: Solids, Fluids, and Reactions

Advanced DI solvers now address generic multiphase, multi-physics environments: inert fluids, elastoplastic solids, and reactive mixtures all described via a single PDE system. Each material is tracked by a volume fraction $\phi_{(l)}$ with local composition/strain/history variables. Mechanical equilibrium and closure relations yield unified velocity, pressure, and stress fields, solving for complex shocks, detonations, and high-rate deformations.

Interface steepening (THINC–BVD), conservative finite-volume updates, and hierarchical AMR deliver near-sharp interfaces with precise mass, momentum, and energy conservation. Benchmarks span detonation–solid interactions, rate-stick, and explosive-metal vessel problems, consistently matching experiments to within 1–2% across regimes and offering AMR scalability to thousands of cores (Wallis et al., 2020).

6. Limitations, Challenges, and Directions

Key limitations of DI-based solvers include the intrinsic interface thickness (ensuring some non-zero mixing), increased numerical diffusion in highly dynamic geometries, and, for DDPM-based models, high computational cost due to the large number of denoising steps needed for sample generation (Luo et al., 2024, Yang et al., 2023). Imposing hard physical constraints, such as incompressibility or mass conservation, within neural diffusion frameworks remains an open challenge, manifesting as non-physical artifacts at long evaluation times.

Proposed extensions include model-physics coupling (embedding PDE residuals and divergence-free projections in network loss), DDPM sampling acceleration (e.g., DDIM, probability flow ODEs), and higher-dimensional/multi-phase applications. On the numerical side, improved discrete conservation, path-conservative fluxes, and interface sharpening continue to be critical in large-deformation and shock-capturing contexts, especially when handling elastic or elastoplastic solids.

7. Applications and Empirical Impact

DiffFluid frameworks are central to a broad class of scientific and engineering problems requiring robust modeling of unsteady, multi-material, and/or optimization-centric fluid systems. They underpin state-of-the-art in:

• Differentiable robot fluid manipulation pipelines (FluidLab, blending MPMs, grid solvers, autodiff, and domain-specific optimization) (Xian et al., 2023).
• Neural surrogates for high-dimensional fluid-structure interaction prediction and control (DDPM + Transformer, multi-resolution noise image-to-image solvers) (Luo et al., 2024, Gachnang et al., 10 Jul 2025).
• Multi-physics simulation for reactive/solid/fluid/shock problems with strong empirical validation and scalability (THINC–BVD, path-conservative DG, AMR) (Wallis et al., 2020, Kemm et al., 2020).
• Computational benchmarks in canonical flows (oscillating droplets, detonation into solids, airfoil aerodynamic fields, high-Reynolds turbulence), where DI and neural DiffFluid models have demonstrated leading accuracy relative to analytical/experimental or operator neural models.

Collectively, DiffFluid methodologies unify physically-principled multiphase modeling with modern gradient-based and data-driven fluid prediction, enabling high-fidelity simulation and robust optimization across a range of scales and regimes.