Author: GE_at_TUM

Here’s also a talk summarizing our recent work on diffusion models for probabilistic Neural solvers: https://youtu.be/xaWxERImy0g

It covers the whole range: from steady state cases, over time-dependent surrogate models, all the way to integrating differentiable simulations into learning score functions. And here are the three corresponding papers:

• Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models , https://arxiv.org/pdf/2312.05320
• Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation , https://arxiv.org/pdf/2309.01745
• Solving Inverse Physics Problems with Score Matching , https://arxiv.org/pdf/2301.10250.pdf

We’re glad to report that Physical Review E has just accepted our paper on neural solvers for wake flows and boundary layers. You can find the full preprint here https://arxiv.org/abs/2308.04296 , and source code is available at https://github.com/tum-pbs/DiffPhys-CylinderWakeFlow/.

Full paper abstract: This work delineates a hybrid predictive framework configured as a coarse-grained surrogate for reconstructing unsteady fluid flows around multiple cylinders of diverse configurations. The presence of cylinders of arbitrary nature causes abrupt changes in the local flow profile while globally exhibiting a wide spectrum of dynamical wakes fluctuating in either a periodic or chaotic manner. Consequently, the focal point of the present study is to establish predictive frameworks that accurately reconstruct the overall fluid velocity flowfield such that the local boundary layer profile, as well as the wake dynamics, are both preserved for long time horizons. The hybrid framework is realized using a base differentiable flow solver combined with a neural network, yielding a differentiable physics-assisted neural network (DPNN). The framework is trained using bodies with arbitrary shapes, and then it is tested and further assessed on out-of-distribution samples. Our results indicate that the neural network acts as a forcing function to correct the local boundary layer profile while also remarkably improving the dissipative nature of the flowfields. It is found that the DPNN framework clearly outperforms the supervised learning approach while respecting the reduced feature space dynamics. The model predictions for arbitrary bodies indicate that the Strouhal number distribution with respect to spacing ratio exhibits similar patterns with existing literature. In addition, our model predictions also enable us to discover similar wake categories for flow past arbitrary bodies. For the chaotic wakes, the present approach predicts the chaotic switch in gap flows up to the mid-time range.

Our work on transonic flows over aerofoils is finally online: https://arxiv.org/abs/2403.17131

Apart from being accurate & stable. We also realized the trained models allow for efficiently performing a “classic” global instability analysis.

It was great to see the networks “really” learn the fundamental physics: When applied to a previously unseen mean flow, the eigenvalue spectrum of the trained network reliably captures the key modes that we’d expect from a transient case.

The unstable modes near 10^-1 along y (with positive real part) indicate the onset of buffet, i.e. growing large scale instabilities in the flow. The smaller modes around 2*10^0 are typical vortex shedding frequencies from the Kelvin-Helmholtz instability.

Full paper abstract: Effectively predicting transonic unsteady flow over an aerofoil poses inherent challenges. In this study, we harness the power of deep neural network (DNN) models using the attention U-Net architecture. Through efficient training of these models, we achieve the capability to capture the complexities of transonic and unsteady flow dynamics at high resolution, even when faced with previously unseen conditions. We demonstrate that by leveraging the differentiability inherent in neural network representations, our approach provides a framework for assessing fundamental physical properties via global instability analysis. This integration bridges deep neural network models and traditional modal analysis, offering valuable insights into transonic flow dynamics and enhancing the interpretability of neural network models in flowfield diagnostics.

For TUM students: we also have a new thesis topic on visual flow capture with RPIs. This is motivated by a previous project: ScalarFlow. There we created a first large-scale data set of reconstructions of real-world smoke plumes. It used an accurate physics-based reconstruction from a small number of video streams. Central components of our framework were a novel estimation of unseen inflow regions and an efficient optimization scheme constrained by a simulation to capture real-world fluids. The published data set contains volumetric reconstructions of velocity and density as well as the corresponding input image sequences.

More info can be found here: https://ge.in.tum.de/download/thesis-MA-rpi.pdf

If you have experience with hardware setups and cameras, please contact us via email at i15ge@cs.tum.edu . We have thesis and or HiWi positions available in this area!

It’s worth pointing out that our paper on “How Temporal Unrolling Supports Neural Physics Simulators” is online now: https://arxiv.org/abs/2402.12971

One of the key findings is: don’t throw away your simulator (yet) even if it doesn’t have gradients. You can get substantial accuracy boosts (around 5x) by coupling an NN with the simulator, instead of letting the NN do all the work… Here are results for a KS and a turbulence case:

Paper Abstract: Unrolling training trajectories over time strongly influences the inference accuracy of neural network-augmented physics simulators. We analyze these effects by studying three variants of training neural networks on discrete ground truth trajectories. In addition to commonly used one-step setups and fully differentiable unrolling, we include a third, less widely used variant: unrolling without temporal gradients. Comparing networks trained with these three modalities makes it possible to disentangle the two dominant effects of unrolling, training distribution shift and long-term gradients. We present a detailed study across physical systems, network sizes, network architectures, training setups, and test scenarios. It provides an empirical basis for our main findings: A non-differentiable but unrolled training setup supported by a numerical solver can yield 4.5-fold improvements over a fully differentiable prediction setup that does not utilize this solver. We also quantify a difference in the accuracy of models trained in a fully differentiable setup compared to their non-differentiable counterparts. While differentiable setups perform best, the accuracy of unrolling without temporal gradients comes comparatively close. Furthermore, we empirically show that these behaviors are invariant to changes in the underlying physical system, the network architecture and size, and the numerical scheme. These results motivate integrating non-differentiable numerical simulators into training setups even if full differentiability is unavailable. We also observe that the convergence rate of common neural architectures is low compared to numerical algorithms. This encourages the use of hybrid approaches combining neural and numerical algorithms to utilize the benefits of both.

The differentiable simulation library Φ_ML (Phi-ML), which is e.g. the basis for projects like PhiFlow, has been accepted in JOSS now! Congratulations Philipp 😀 👍 The full version is available here: https://joss.theoj.org/papers/10.21105/joss.06171

Short summary: Φ_ML is a math and neural network library designed for science applications. It enables you to quickly evaluate many network architectures on your data sets, perform linear and non-linear optimization, and write differentiable simulations. Φ_ML is compatible with Jax, PyTorch, TensorFlow and NumPy and your code can be executed on all of these backends.

We’re happy to report two accepted papers at ICLR 2024! Congrats Patrick and Rene 😀 👍 They’re on particle-based learning https://openreview.net/forum?id=HKgRwNhI9R and stabilized backprop through time https://openreview.net/forum?id=bozbTTWcaw, additional details, code etc. will follow soon. For now here are the two abstracts in full:

Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics: Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. 

Stabilizing Backpropagation Through Time to Learn Complex Physics: Of all the vector fields surrounding the minima of recurrent learning setups, the gradient field with its exploding and vanishing updates appears a poor choice for optimization, offering little beyond efficient computability. We seek to improve this suboptimal practice in the context of physics simulations, where backpropagating feedback through many unrolled time steps is considered crucial to acquiring temporally coherent behavior. The alternative vector field we propose follows from two principles: physics simulators, unlike neural networks, have a balanced gradient flow and certain modifications to the backpropagation pass leave the positions of the original minima unchanged. As any modification of backpropagation decouples forward and backward pass, the rotation-free character of the gradient field is lost. Therefore, we discuss the negative implications of using such a rotational vector field for optimization and how to counteract them. Our final procedure is easily implementable via a sequence of gradient stopping and component-wise comparison operations, which do not negatively affect scalability. Our experiments on three control problems show that especially as we increase the complexity of each task, the unbalanced updates from the gradient can no longer provide the precise control signals necessary while our method still solves the tasks.

Interested in trying out diffusion-based “neural simulators” for fluids yourselves? We’ve just added a notebook that let’s you get started with training and probabilistic inference right away: https://github.com/tum-pbs/autoreg-pde-diffusion/blob/main/acdm-demo.ipynb

The image above shows a few generated posterior samples for the (tough) transonic flow dataset. Alternatively, you can also directly run it in colab via this link: https://colab.research.google.com/github/tum-pbs/autoreg-pde-diffusion/blob/main/acdm-demo.ipynb

Note that this model loads a pre-trained diffusion model, and runs fine-tuning for 10 epochs. The full training would require ca. one day of runtime. Here’s also the temporal evaluation from the notebook:

We finally also have the source code for our RB-control paper online: https://github.com/tum-pbs/two-way-coupled-control , the paper being available here: https://ge.in.tum.de/download/two-way-control.pdf.

A differentiable flow solver is used to train a controller that steers a rigid body to reach a goal position and orientation. Interestingly, the differentiable solver learns much faster and more reliably than the reinforcement learning variants we tried, and it clearly outperforms simpler baselines:

Paper abstract: We investigate the use of deep neural networks to control complex nonlinear dynamical systems, specifically the movement of a rigid body immersed in a fluid. We solve the Navier Stokes equations with two way coupling, which gives rise to nonlinear perturbations that make the control task very challenging. Neural networks are trained in an unsupervised way to act as controllers with desired characteristics through a process of learning from a differentiable simulator. Here we introduce a set of physically interpretable loss terms to let the networks learn robust and stable interactions. We demonstrate that controllers trained in a canonical setting with quiescent initial conditions reliably generalize to varied and challenging environments such as previously unseen inflow conditions and forcing, although they do not have any fluid information as input. Further, we show that controllers trained with our approach outperform a variety of classical and learned alternatives in terms of evaluation metrics and generalization capabilities.

The source code for our turbulent flow simulations using Autoregressive Conditional Diffusion Models (ACDMs) is online now at https://github.com/tum-pbs/autoreg-pde-diffusion , let us know how it works!

Project summary: Our work targets the prediction of turbulent flow fields from an initial condition using autoregressive conditional diffusion models (ACDMs). Our method relies on the DDPM approach, a class of generative models based on a parameterized Markov chain. They can be trained to learn the conditional distribution of a target variable given a conditioning. In our case, the target variable is the flow field at the next time step, and the conditioning is the flow field at the current time step, i.e., the simulation trajectory is created via autoregressive unrolling of the model. We showed that ACDMs can accurately and probabilistically predict turbulent flow fields, and that the resulting trajectories align with the statistics of the underlying physics. Furthermore, ACDMs can generalize to flow parameters beyond the training regime, and exhibit high temporal rollout stability, without compromising the quality of generated samples.

More details can also be found on the project website.

We’re happy to report that our paper on learning hybrid solvers for reactive flows has now been accepted at the Data-Centric Engineering journal! The source code is also available now: https://github.com/tum-pbs/Hybrid-Solver-for-Reactive-Flows

Full paper abstract: Modeling complex dynamical systems with only partial knowledge of their physical mechanisms is a crucial problem across all scientific and engineering disciplines. Purely data-driven approaches, which only make use of an artificial neural network and data, often fail to accurately simulate the evolution of the system dynamics over a sufficiently long time and in a physically consistent manner. Therefore, we propose a hybrid approach that uses a neural network model in combination with an incomplete partial differential equations (PDE) solver that provides known, but incomplete physical information. In this study, we demonstrate that the results obtained from the incomplete PDEs can be efficiently corrected at every time step by the proposed hybrid neural network – PDE solver model, so that the effect of the unknown physics present in the system is correctly accounted for. For validation purposes, the obtained simulations of the hybrid model are successfully compared against results coming from the complete set of PDEs describing the full physics of the considered system. We demonstrate the validity of the proposed approach on a reactive flow, an archetypal multi-physics system that combines fluid mechanics and chemistry, the latter being the physics considered unknown. Experiments are made on planar and Bunsen-type flames at various operating conditions. The hybrid neural network – PDE approach correctly models the flame evolution of the cases under study for significantly long time windows, yields improved generalization, and allows for larger simulation time steps.

Our paper on autoregressive diffusion models for improved temporal predictions of complex PDE-based simulations is online now at https://arxiv.org/abs/2309.01745 (ACDM). Above you can see a preview output from a transonic turbulent flow case. Especially the shock waves around the obstacle are very tough here.

To summarize the results of ACDM: it’s highly accurate and turns a regular Neural PDE solver into a probabilistic method, i.e. it can compute different versions of the solution via posterior sampling. This is an example:

A key insight is also that the diffusion training (ACDM) yields an excellent temporal stability, despite being trained without any unrolling. Here’s a preview ACDM in orange versus a few baselines (GT in black):

Full abstract: Simulating turbulent flows is crucial for a wide range of applications, and machine learning-based solvers are gaining increasing relevance. However, achieving stability when generalizing to longer rollout horizons remains a persistent challenge for learned PDE solvers. We address this challenge by introducing a fully data-driven fluid solver that utilizes an autoregressive rollout based on conditional diffusion models. We show that this approach offers clear advantages in terms of rollout stability compared to other learned baselines. Remarkably, these improvements in stability are achieved without compromising the quality of generated samples, and our model successfully generalizes to flow parameters beyond the training regime. Additionally, the probabilistic nature of the diffusion approach allows for inferring predictions that align with the statistics of the underlying physics. We quantitatively and qualitatively evaluate the performance of our method on a range of challenging scenarios, including incompressible Navier-Stokes and transonic flows, as well as isotropic turbulence.

The project page can be found here.