Physics-integrated neural differentiable modeling for immersed boundary systems
Authors: Chenglin Li, Hang Xu, Jianting Chen, Yanfei Zhang
Organizations: State Key Lab of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · State Key Laboratory of Maritime Technology and Safety, Shanghai Ship and Shipping Research Institute Co., Ltd, Shanghai 200135, China.
Accurately, efficiently, and stably computing complex fluid flows and their evolution near solid boundaries over long horizons remains challenging. Conventional numerical solvers require fine grids and small time steps to resolve near-wall dynamics, resulting in high computational costs, while purely data-driven surrogate models accumulate rollout errors and lack robustness under extrapolative conditions. To address these issues, this study extends existing neural PDE solvers by developing a physics-integrated differentiable framework for long-horizon prediction of immersed-boundary flows. A key design aspect of the framework includes an important improvement, namely the structural integration of physical principles into an end-to-end differentiable architecture incorporating a PDE-based intermediate velocity module and a multi-direct forcing immersed boundary module, both adhering to the pressure-projection procedure for incompressible flow computation. The computationally expensive pressure projection step is substituted with a learned implicit correction using ConvResNet blocks to reduce cost, and a sub-iteration strategy is introduced to separate the embedded physics module's stability requirement from the surrogate model's time step, enabling stable coarse-grid autoregressive rollouts with large effective time increments. The framework uses only single-step supervision for training, eliminating long-horizon backpropagation and reducing training time to under one hour on a single GPU. Evaluations on benchmark cases of flow past a stationary cylinder and a rotationally oscillating cylinder at Re=100 show the proposed model consistently outperforms purely data-driven, physics-loss-constrained, and coarse-grid numerical baselines in flow-field fidelity and long-horizon stability, while achieving an approximately 200-fold inference speedup over the high-resolution solver.
Figures & tables
Figure 1: Schematic of the computational setup and domains. The immersed-boundary (IB) solver is defined on the global fluid domain Ωf (gray box) with boundaries Γinlet , Γoutlet , Γupper , and Γlower . The surrogate model is evaluated on a truncated subdomain Ωfr (blue box) bounded by Γinletr , Γoutletr , Γupperr , and Γlowerr . A prescribed inflow u=uin is imposed at Γinlet , and free-slip conditions are enforced on Γupper and Γlower .
Figure 2: Schematic of the proposed physics-integrated differentiable neural network. Trainable neural modules (ConvResNet blocks) are shown in green, while non-trainable physics operators (the PDE operator and the multi-direct-forcing IBM module) are highlighted in yellow. Grey dashed boxes enclose the inputs, outputs, and intermediate physical variables. Data propagate along the black arrows. The PDE-operator substep is iterated N times, and up-/down-sampling bridges the resolutions between the physics and neural components.
Figure 3: Comparison of spatio-temporal vorticity predictions by different neural models for flow past a stationary circular cylinder at (Re=100,ω=0) .
Figure 4: Comparison of structural force predictions by different neural models for flow past a stationary circular cylinder at (Re=100,ω=0) . The left column shows the time histories of the drag coefficient CD , and the right column shows the lift coefficient CL . The gray-shaded interval denotes the training window, whereas the white interval corresponds to out-of-distribution time extrapolation beyond the training horizon. The black solid line represents the ground truth, and the blue curve corresponds to the prediction of the proposed model.
Figure 5: Stepwise rollout relative error ϵt for flow past a stationary circular cylinder at Re=100 and ω=0 , defined in Eq. ( 25 ). The metric is computed as the average of the relative L2 errors of u , v , and the structural response vector F at each rollout step. The gray-shaded region indicates the training window, and the white region indicates temporal extrapolation beyond the training horizon.
Figure 6: Comparison of spatio-temporal vorticity predictions by different neural models for flow past a rotationally oscillating circular cylinder at (Re=100,ω=4) .
Figure 7: Comparison of structural force predictions by different neural models for flow past a rotationally oscillating circular cylinder at (Re=100,ω=4) . The left column shows the time histories of the drag coefficient CD , and the right column shows the lift coefficient CL . The gray-shaded interval denotes the training window, whereas the white interval corresponds to out-of-distribution time extrapolation beyond the training horizon. The black solid line represents the ground truth, and the blue curve corresponds to the prediction of the proposed model.
Figure 8: Stepwise rollout relative error ϵt for flow past a rotationally oscillating circular cylinder at Re=100 and ω=4 , defined in Eq. ( 25 ). The metric is computed as the average of the relative L2 errors of u , v , and the structural response vector F at each rollout step. The gray-shaded region indicates the training window, and the white region indicates temporal extrapolation beyond the training horizon.
Figure 9: Average relative error ϵˉ (averaged over the entire inference horizon) under unseen operating conditions for the rotationally oscillating cylinder. Models are trained on ω={2,4} and evaluated at ω=3 (interpolation) and ω=5 (extrapolation). Lower values indicate better long-horizon generalization.
Figure 10: Comparison of spatio-temporal vorticity predictions by different neural models for flow past a rotationally oscillating circular cylinder at (Re=100,ω=5) .
Figure 11: Comparison of structural force predictions by different neural models for flow past a rotationally oscillating circular cylinder at (Re=100,ω=5) . The left column shows the time histories of the drag coefficient CD , and the right column shows the lift coefficient CL . The gray-shaded interval denotes the training window, whereas the white interval corresponds to out-of-distribution time extrapolation beyond the training horizon. The black solid line represents the ground truth, and the blue curve corresponds to the prediction of the proposed model.
Figure 12: Wall-clock time per learning step for different methods. The neural models are evaluated on the truncated subdomain Ωfr using an 80×40 grid, whereas the ground-truth CFD solution is computed on the full domain Ωf using a 717×382 grid.
Figure 13: Training efficiency comparison between the single-step and BPTT-based multi-step training for flow past a stationary circular cylinder at Reynolds number Re=100 . The shaded bands represent ±σ .
Training strategy
GPU memory usage (GB)
Single-step training
2.7
BPTT ( T=3 )
5.7
BPTT ( T=5 )
8.8
Table 1: GPU memory consumption under different training strategies.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
CˉD
CL′
St
Our
1.383
±0.345
0.167
Cui et al. [ 47 ]
1.360
±0.340
0.167
Uhlmann [ 48 ]
1.453
±0.339
0.169
Wang et al. [ 49 ]
1.334
±0.370
0.163
Appendix
Table A1: Dimensionless coefficients obtained from the simulation of the flow around a stationary cylinder at Re=100 , using D=1 and dt=0.005 .
Figure B1: Illustration of transferring staggered-grid variables from a fine mesh to an n×n coarser mesh via downsampling (an 8× reduction in resolution is used in this study). For each coarse control volume, the face-centered velocity components on the fine grid are area-averaged to obtain the corresponding coarse-grid velocities, while the cell-centered pressures within the fine-grid subcells are averaged to yield the coarse-grid pressure. Fine-grid velocity samples located in the interior of the coarse cell are not retained.
Figure C2: Architectures of the baseline models used for comparison. (a) Purely data-driven model trained with the data loss Lr . (b) Physics loss constrained model, which augments (a) with the physics-consistency term Lphy=LIB+Lm+Ldiv and optimizes the total loss Ltotal=αLphy+βLr .
Fluid flows are governed by the nonlinear Navier-Stokes equations, which can manifest multiscale dynamics even from predictable initial conditions. Predicting such phenomena remains a formidable challenge in scientific machine learning, particularly regarding convergence speed, data requirements, and solution accuracy. In complex fluid flows, these challenges are exacerbated by long-range spatial dependencies arising from distant boundary conditions, which typically necessitate extensive supervision data to achieve acceptable results. We propose the Domain-Decomposed and Shifted Physics-Informed Neural Network (DDS-PINN), a framework designed to resolve such multiscale interactions with minimal supervision. By utilizing localized networks with a unified global loss, DDS-PINN captures global dependencies while maintaining local precision. The robustness of the approach is demonstrated across a suite of benchmarks, including a multiscale linear differential equation, the nonlinear Burgers' equation, and data-free Navier-Stokes simulations of flat-plate boundary layers. Finally, DDS-PINN is applied to the computationally challenging backward-facing step (BFS) problem; for laminar regimes (Re = 100), the model yields results comparable to computational fluid dynamics (CFD) without the need for any data, accurately predicting boundary layer thickness, separation, and reattachment lengths. For turbulent BFS flow at Re = 10,000, the framework achieves convergence to O(10^-4) using only 500 random supervision points (< 0.3 % of the total domain), outperforming established methods like Residual-based Attention-PINN in accuracy. This approach demonstrates strong potential for the super-resolution of complex turbulent flows from sparse experimental measurements.
Prashant Kumar, Rajesh Ranjan
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016
This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system R(w)=0, with a generic additive, parametrized, and differentiable correction fφ(w), with trainable parameters φ. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.
Predicting complex spatiotemporal dynamics in physical processes often demands computationally expensive numerical methods or data-driven neural networks that suffer from high training costs, error accumulation, and limited generalizability to unseen parameters. An effective approach to address these challenges is leveraging physics priors in training neural networks, known as physics-informed deep learning (PiDL). In this work, we introduce the Multi-Resolution Finite-Volume-inspired network, MuRFiV, designed to capitalize on the conservative property of finite volume on the global scale and the expressive power of deep learning on the local scale. We demonstrate the effectiveness of MuRFiV on several spatio-temporal systems governed by partial differential equations (PDEs), including Burgers' equation, shallow water equations, and incompressible Navier-Stokes equations. By embedding PDE information into the deep learning architecture, MuRFiV achieves strong long-term prediction accuracy and remains stable over very long autoregressive rollouts, significantly outperforming data-driven neural network baselines. This result highlights the promise of combining multiresolution learning with finite-volume-inspired inductive bias for accurate and robust long-term prediction of complex dynamics.
Xin-Yang Liu, Xiantao Fan, Jian-Xun Wang
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA · Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA