We have also just released the PDF Version of our Physics-based Deep Learning book. It can come in handy if you want to read it offline in one go: https://arxiv.org/pdf/2109.05237.pdf

Looking ahead, we’re of course also planning a printed version with extra-thick, glossy paper and a large font size so that typing up the Jupyter notebooks is extra convenient. (No, of course not đ – rather, please don’t print the PDF!)

In addition, there’s also an introductory video that summarizes the goals of the PBDL book, and runs through some of the highlights: https://youtu.be/SU-OILSmR1M

Its central goal is to give a thorough, hands-on introduction to deep learning for physical systems, from simple physical losses to full hybrid solvers. Best of all: the majority of the topics come with Jupyter notebooks that can be run on the spot!

The PBDL book contains a practical and comprehensive introduction of everything related to deep learning in the context of physical simulations. As much as possible, all topics come with hands-on code examples in the form of Jupyter notebooks to quickly get started. Beyond standard supervised learning from data, weâll look at physical loss constraints, more tightly coupled learning algorithms with differentiable simulations, as well as reinforcement learning and uncertainty modeling. We live in exciting times: these methods have a huge potential to fundamentally change what we can achieve with simulations.

The key aspects that we will address in the following are:

explain how to use deep learning techniques to solve PDE problems,

how to combine them with existing knowledge of physics,

without discarding our knowledge about numerical methods.

The focus of this book lies on:

Field-based simulations (no Lagrangian methods)

Combinations with deep learning (plenty of other interesting ML techniques, but not here)

Experiments as outlook (i.e., replace synthetic data with real-world observations)

The name of this book, Physics-Based Deep Learning, denotes combinations of physical modeling and numerical simulations with methods based on artificial neural networks. The general direction of Physics-Based Deep Learning represents a very active, quickly growing and exciting field of research.

The aim is to build on all the powerful numerical techniques that we have at our disposal, and use them wherever we can. As such, a central goal of this book is to reconcile the data-centered viewpoint with physical simulations.

The resulting methods have a huge potential to improve what can be done with numerical methods: in scenarios where a solver targets cases from a certain well-defined problem domain repeatedly, it can for instance make a lot of sense to once invest significant resources to train a neural network that supports the repeated solves. Based on the domain-specific specialization of this network, such a hybrid could vastly outperform traditional, generic solvers.

Our paper on high-accuracy airfoil flow predictions (Reynolds-averaged Navier-Stokes) with deep neural networks is online now. Interestingly, turns out you don’t need things like complex graph neural networks to handle adaptive / changing meshes. The preprint is available on arXiv: https://arxiv.org/abs/2109.02183

Even the “worst case” results are very accurate, here’s an example comparison:

The trained neural network yields results that resolve all necessary structures such as shocks, and has an average error of less than 0.3% for turbulent transonic cases. This is appropriate for real-world, industrial applications. Here’s are the inferred skin friction coefficent for ‘e221’:

We also directly compare to a graph neural network (GNN) approach (one that is additionally coupled with a solver), yielding a ca. 3.6x improvement in terms of RMSE across all inferred fields. This is achieved with a much simpler and faster method…

Full Paper Abstract: The present study investigates the accurate inference of Reynolds-averaged Navier-Stokes solutions for the compressible flow over aerofoils in two dimensions with a deep neural network. Our approach yields networks that learn to generate precise flow fields for varying body-fitted, structured grids by providing them with an encoding of the corresponding mapping to a canonical space for the solutions. We apply the deep neural network model to a benchmark case of incompressible flow at randomly given angles of attack and Reynolds numbers and achieve an improvement of more than an order of magnitude compared to previous work. Further, for transonic flow cases, the deep neural network model accurately predicts complex flow behaviour at high Reynolds numbers, such as shock wave/boundary layer interaction, and quantitative distributions like pressure coefficient, skin friction coefficient as well as wake total pressure profiles downstream of aerofoils. The proposed deep learning method significantly speeds up the predictions of flow fields and shows promise for enabling fast aerodynamic designs.

The learning process is enabled (among others) by a discriminator that self-supervises in terms of a physical quantity such as the marker density.

That makes it possible to obtain velocities and run a simulations purely based on a marker density. We actually couldn’t rewrite the NS equations in this way, but the generator learns it from the training data.

Full paper abstract: While modern fluid simulation methods achieve high-quality simulation results, it is still a big challenge to interpret and control motion from visual quantities, such as the advected marker density. These visual quantities play an important role in user interactions: Being familiar and meaningful to humans, these quantities have a strong correlation with the underlying motion. We propose a novel data-driven conditional adversarial model that solves the challenging, and theoretically ill-posed problem of deriving plausible velocity fields from a single frame of a density field. Besides density modifications, our generative model is the first to enable the control of the results using all of the following control modalities: obstacles, physical parameters, kinetic energy, and vorticity. Our method is based on a new conditional generative adversarial neural network that explicitly embeds physical quantities into the learned latent space, and a new cyclic adversarial network design for control disentanglement. We show the high quality and versatile controllability of our results for density-based inference, realistic obstacle interaction, and sensitive responses to modifications of physical parameters, kinetic energy, and vorticity.

It contains everything’s that necessary to train a neural network to perform super fast shape optimizations to reduce the drag of a level-set based shape immersed in a moving fluid. The neural network is inherently differentiable, and very fast to evaluate. Hence, the trained networks represent a great building block for inverse problems. For completeness, here’s the full abstract of the paper.

Abstract: Efficiently predicting the flowfield and load in aerodynamic shape optimisation remains a highly challenging and relevant task. Deep learning methods have been of particular interest for such problems, due to their success for solving inverse problems in other fields. In the present study, U-net based deep neural network (DNN) models are trained with high-fidelity datasets to infer flow fields, and then employed as surrogate models to carry out the shape optimisation problem, i.e. to find a drag minimal profile with a fixed cross-section area subjected to a two-dimensional steady laminar flow. A level-set method as well as Bezier-curve method are used to parameterise the shape, while trained neural networks inÂ conjunction with automatic differentiation are utilized to calculate the gradient flow in the optimisation framework. The optimised shapes and drag force values calculated from the flowfields predicted by DNN models agree well with reference data obtained via a Navier-Stokes solver and from the literature, which demonstrates that the DNN models are capable of predicting not only flowfield but also yield satisfactory aerodynamic forces. This is particularly promising as the DNNs were not specifically trained to infer aerodynamic forces. In conjunction with the fast runtime, the DNN-based optimisation framework shows promise for general aerodynamic design problems.

We’re happy to report that all three CVPR papers are online now. They cover a wide range of topics, from differentiable physics and rendering (for fluids), over learning collision free spaces (for cloth) to dynamics scenes (for neural rendering).

Our own entry in the WeatherBench benchmark is now published in the Journal of Advances in Modeling Earth Systems. It outperforms existing works with an RMSE of 268 and 499 for 3 and 5 day Z500 forecasts, respectively. It’s also at least on-par with a full traditional model running at a similar resolution. That being said – it’s still clearly falling behind the operational forecasting reference. Hopefully, it will inspire more people to join the WeatherBench challenge, and further improve the forecasts!

Paper Abstract: Numerical weather prediction has traditionally been based on the models that discretize the dynamical and physical equations of the atmosphere. Recently, however, the rise of deep learning has created increased interest in purely dataâdriven mediumârange weather forecasting with first studies exploring the feasibility of such an approach. To accelerate progress in this area, the WeatherBench benchmark challenge was defined. Here, we train a deep residual convolutional neural network (Resnet) to predict geopotential, temperature and precipitation at 5.625Â° resolution up to 5 days ahead. To avoid overfitting and improve forecast skill, we pretrain the model using historical climate model output before fineâtuning on reanalysis data. The resulting forecasts outperform previous submissions to WeatherBench and are comparable in skill to a physical baseline at similar resolution. We also analyze how the neural network makes its predictions and find that the model has learned reasonable physically reasonable correlations.

The nice image there is from our temporally-coherent fluid GAN (tempoGAN), published in 2018 at SIGGRAPH. Interestingly, since then few works were able to handle 4D data sets (3D volumes over time) while taking into account how the learned functions should change over time.

The cutout above is from our largest example, with a resolution of 1024 Ă 720 Ă 720 cells over 200 time steps. That means the CNN generated a total number of 6,794,772,480,000 cells (i.e., more than 6 trillion cells) for this sequence.

We’re happy to report that three of our papers have been accepted to the CVPR 2021 conference, two of them being orals. Details will follow in the next weeks, but as a preview we have:

Global Transport for Fluid Reconstruction with Learned Self-Supervision (oral), together with the CGL at ETH Zurich, congratulations Erik!

Neural Scene Graphs for Dynamic Scenes (oral), together with AlgoLux and the Princeton CI lab, congratulations Julian!

Self-Supervised Collision Handling via Generative 3D Garment Models for Virtual Try-On, together with the with the Multimodal Simulation Lab at Universidad Rey Juan Carlos, congratulations Igor!

Nils Thuerey recently gave a talk at the LLNL (https://www.llnl.gov/) about Differentiable Physics Simulations for Deep Learning. While we’re preparing the next release of our differentiable simulation framework PhiFlow (https://github.com/tum-pbs/PhiFlow), you can check out the talk here:

Talk abstract: In this talk I will focus on the possibilities that arise from recent advances in the area of deep learning for physical simulations. In this context, especially the Navier-Stokes equations represent an interesting and challenging advection-diffusion PDE that poses a variety of challenges for deep learning methods.

In particular, I will focus on differentiable physics solvers within the larger field of differentiable programming. Differentiable solvers are very powerful tools to guide deep learning processes, and support finding desirable solutions. The existing numerical methods for efficient solvers can be leveraged within learning tasks to provide crucial information in the form of reliable gradients to update the weights of a neural networks. Interestingly, it turns out to be beneficial to combine supervised and physics-based approaches. The former poses a much simpler learning task by providing explicit reference data that is typically pre-computed. Physics-based learning on the other hand can provide gradients for a larger space of states that are only encountered during training runs. Here, differentiable solvers are particularly powerful to, e.g., provide neural networks with feedback about how inferred solutions influence the long-term behavior of a physical model.

I will demonstrate this concept with several examples from learning to reduce numerical errors, over long-term planning and control, to generalization. I will conclude by discussing current limitations and by giving an outlook about promising future directions.

Despite being a challenging year due to various non-research related reasons (Covid, anyone?), the TUM P.B.S. group can celebrate a very successful year. We’ve had a very nice series of publications, among others with papers at the NeurIPS, ICML and ICLR conferences.

Our Paper on deep learning algorithms interacting with differentiable PDE solvers was just successfully presented at NeurIPS. And just in time for the conference, we also finished uploading the last piece of the corresponding source code release.

This is the full abstract of the paper: Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby allow the model to interact with the PDE during training. This provides the model with realistic input distributions that take previous corrections into account, yielding improvements in accuracy with stable rollouts of several hundred recurrent evaluation steps and surpassing even tailored supervised variants. We highlight the performance of the differentiable physics networks for a wide variety of PDEs, from non-linear advection-diffusion systems to three-dimensional Navier-Stokes flows.

Additional details can be found on the project page.

Title: Differentiable Physics Simulations for Deep Learning Algorithms

Abstract: Differentiable physics solvers (from the broader field of differentiable programming) show particular promise for including prior knowledge into machine learning algorithms. Differentiable operators were shown to be powerful tools to guide deep learning processes, and PDEs provide a wide range of components to build such operators. They also represent a natural way for traditional solvers and deep learning methods to coexist: Using PDE solvers as differentiable operators in neural networks allows us to leverage existing numerical methods for efficient solvers, e.g., to provide reliable and flexible gradients to update the weights during a learning run.

Interestingly, it turns out to be beneficial to combine âtraditionalâ supervised and physics-based approaches. The former poses a much more straightforward and more stable learning task by providing explicit reference data, while physics-based learning can provide gradients for a larger space of states that are only encountered at training time. Here, differentiable solvers are particularly powerful, e.g., to provide neural networks with feedback about how inferred solutions influence a physical modelâs long-term behavior. I will show and discuss examples with various advection-diffusion type PDEs, among others the Navier-Stokes equations for fluids, for different learning applications. These demonstrations will highlight the properties and capabilities of PDE-powered deep neural networks and serve as a starting point for discussing future developments.

In addition to our paper at the NeurIPS 2020 main conference (which targets deep learning via differentiable PDE solvers for numerical error reduction) we are excited about contributions to the following four NeurIPS workshops. Details will follow over the course of the next weeks, but these workshops very nicely align with our goals to fuse deep learning, numerical methods and physical simulations as seamlessly as possible. E.g., we will present our work on shape optimizations for Navier-Stokes flows as well as our differentiable physics framework phiflow.

For now, we can highly recommend checking out the workshops themselves:

Differentiable Vision, Graphics, and Physics in Machine Learning http://montrealrobotics.ca/diffcvgp/ Organizers: Krishna Jatavallabhula , Kelsey Allen , Victoria Dean , Johanna Hansen , Shuran Song , Florian Shkurti , Liam Paull , Derek Nowrouzezahrai , Josh Tenenbaum

Interpretable Inductive Biases and Physically Structured Learning https://inductive-biases.github.io/ Organizers: Shirley Ho , Michael Lutter , Alexander Terenin , Lei Wang

Machine Learning for Engineering Modeling, Simulation, and Design https://ml4eng.github.io/ Organizers: Alex Beatson , Priya L. Donti , Amira Abdel-Rahman , Stephan Hoyer , Rose Yu , J. Zico Kolter , Ryan P. Adam

Machine Learning and the Physical Sciences https://ml4physicalsciences.github.io/2020/ Organizers: AtÄ±lÄ±m GĂŒneĆ Baydin , Juan Felipe Carrasquilla , Adji Bousso Dieng , Karthik Kashinath , Gilles Louppe , Brian Nord , Michela Paganini , Savannah Thais

Our paper “Numerical investigation of minimum drag profiles in laminar flow using deep learning surrogates” is online now as a preprint. It targets optimizing shapes by using trained deep neural networks models that infer manifolds of Navier-Stokes solutions. We evaluate accuracy and performance of using pretrained models to minimize the drag for shapes immersed in a moving fluid in low Reynolds number regimes.

Full abstract: Efficiently predicting the flowfield and load in aerodynamic shape optimisation remains a highly challenging and relevant task. Deep learning methods have been of particular interest for such problems, due to their success for solving inverse problems in other fields. In the present study, U-net based deep neural network (DNN) models are trained with high-fidelity datasets to infer flow fields, and then employed as surrogate models to carry out the shape optimisation problem, i.e. to find a drag minimal profile with a fixed cross-section area subjected to a two-dimensional steady laminar flow. A level-set method as well as B{\’e}zier-curve method are used to parameterise the shape, while trained neural networks in conjunction with automatic differentiation are utilized to calculate the gradient flow in the optimisation framework. The optimised shapes and drag force values calculated from the flowfields predicted by DNN models agree well with reference data obtained via a Navier-Stokes solver and from the literature, which demonstrates that the DNN models are capable of predicting not only flowfield but also yield satisfactory aerodynamic forces. This is particularly promising as the DNNs were not specifically trained to infer aerodynamic forces. In conjunction with the fast runtime, the DNN-based optimisation framework shows promise for general aerodynamic design problems.

Full abstract: We propose an end-to-end trained neural network architecture to robustly predict the complex dynamics of fluid flows with high temporal stability. We focus on single-phase smoke simulations in 2D and 3D based on the incompressible Navier-Stokes (NS) equations, which are relevant for a wide range of practical problems. To achieve stable predictions for long-term flow sequences, a convolutional neural network (CNN) is trained for spatial compression in combination with a temporal prediction network that consists of stacked Long Short-Term Memory (LSTM) layers. Our core contribution is a novel latent space subdivision (LSS) to separate the respective input quantities into individual parts of the encoded latent space domain. This allows to distinctively alter the encoded quantities without interfering with the remaining latent space values and hence maximizes external control. By selectively overwriting parts of the predicted latent space points, our proposed method is capable to robustly predict long-term sequences of complex physics problems. In addition, we highlight the benefits of a recurrent training on the latent space creation, which is performed by the spatial compression network.

Our paper on improving neural network generalization via a forward-backward pass is also finally online, together with a first code example. A common question that we get about this project: “why racecar“? This is worth explaining here in a bit more detail: it’s not about the speed of the method, but rather racecar is a nice palindrome. Hence, if you reverse the word, you still have “racecar”. Our training approach also makes use of a reversed neural network architecture, re-using all existent building blocks of the network and their weights, somewhat similar to a palindrome. Hence the name. Interestingly, this reverse structure yields an embedding of singular vectors into the weight matrices, and improves performance for new tasks, as we show for a variety of classification and generation tasks in our paper.

Paper Abstract: We propose a novel training approach for improving the generalization in neural networks. We show that in contrast to regular constraints for orthogonality, our approach represents aÂ data-dependentÂ orthogonality constraint, and is closely related to singular value decompositions of the weight matrices. We also show how our formulation is easy to realize in practical network architectures via a reverse pass, which aims for reconstructing the full sequence of internal states of the network. Despite being a surprisingly simple change, we demonstrate that this forward-backward training approach, which we refer to asÂ racecarÂ training, leads to significantly more generic features being extracted from a given data set. Networks trained with our approach show more balanced mutual information between input and output throughout all layers, yield improved explainability and, exhibit improved performance for a variety of tasks and task transfers.

Our new paper “Purely data-driven medium-range weather forecasting achieves comparable skill to physical models at similar resolution” is available now on arXiv: https://arxiv.org/abs/2008.08626

We show that with enough data, a deep-learning based model can actually compete and in some cases outperform established physical models (e.g., IFS forecasts for 210km resolution). We show how such models can be trained based on the WeatherBench data set, that they contain plausible learned structures, and also fare well for challenging fields such precipitation. At the same time, they illustrate that it will be very difficult to increase the performance only with the data that is currently available.

Full abstract: Numerical weather prediction has traditionally been based on physical models of the atmosphere. Recently, however, the rise of deep learning has created increased interest in purely data-driven medium-range weather forecasting with first studies exploring the feasibility of such an approach. Here, we train a significantly larger model than in previous studies to predict geopotential, temperature and precipitation up to 5 days ahead and achieve comparable skill to a physical model run at similar horizontal resolution. Crucially, we pretrain our models on historical climate model output before fine-tuning them on the reanalysis data. We also analyze how the neural network creates its predictions and find that, with some exceptions, it is compatible with physical reasoning. Our results indicate that, given enough training data, data-driven models can compete with physical models. At the same time, there is likely not enough data to scale this approach to the resolutions of current operational models.

Our results demonstrate that Differentiable Physics are a powerful tool, and they neatly fit into the current larger deep learning trend of generic “Differentiable Programming”. They not only yield very good minimizers in terms of well-trained neural networks: a nice side effect is that they allow for leveraging all the existing powerful numerical methods that exist for physical simulations, and employ them to improve training deep neural nets.

Solver-in-the-Loop Paper Abstract: Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby allow the model to interact with the PDE during training. This provides the model with realistic input distributions that take previous corrections into account, yielding improvements in accuracy with stable rollouts of several hundred recurrent evaluation steps and surpassing even tailored supervised variants. We highlight the performance of the differentiable physics networks for a wide variety of PDEs, from non-linear advection-diffusion systems to three-dimensional Navier-Stokes flows.

We’re happy to report that the final version of our paper on “Learning Similarity Metrics for Numerical Simulations” to be presented at the International Conference on Machine Learning (ICML) is online now. We propose learning a metric for data produced by numerical simulations, i.e. PDEs such as Navier-Stokes, and a way to train Siamese networks with a correlation-based loss to improve the inference of similarities. The resulting deep learning based metric outperforms simpler metrics and other learned metrics such as LPIPS.

Assessing similarity for complex data is is a fundamental problem in all computational disciplines ranging from simulations of blood flow to aircraft design. Many practical problems rely on highly complex PDEs, where small perturbations in the input drastically alter the solutions. Regular vector space metrics like the LÂČ distance are unreliable as they perform an element-wise comparison, and thus cannot compare contextual information or structures on different scales. Our approach, dubbed LSiM, employs convolutional neural networks (CNNs) as a method to extract and compare more meaningful features from a set of two simulation frames.