The Unreasonable Ineffectivness of NN-Accuracy Scaling

TL;DR: There’s a surprising aspect of our recent paper that’s easy to overlook: we noticed that NN test error scales sub-optimally with -1/3 over parameter count. This is for correction, while prediction tasks are slightly worse with -1/4. Our results indicate that this is stable across physical systems and network architectures!

To provide more background: many experiments from our paper “How Temporal Unrolling Supports Neural Physics Simulators” show clear, continuous improvements in accuracy for increasing network sizes.

An obvious conclusion is that larger networks achieve better results. However, in the context of scientific computing, simply increasing the network size further and further is not an attractive option. As neural approaches compete with established numerical methods, applying pure neural or hybrid architectures always entails accuracy, efficiency, and scaling considerations. The scaling of networks towards real-world engineering problems on physical systems has been an open question, and overly large networks will be more resource hungry than established solvers in the worst case.

To shed light here, we computed a convergence rate between the test loss and the number of network parameters. We used an average test loss for each individual combination of network size and training setup. The average is computed over the full set of random seeds used in our study, i.e., 8 to 20 individual training runs per size and variant depending on the physical system (more than 800 models for the 3 graphs above). For the correction setups (NN+coarse solver), we estimate the convergence rate of the correction networks with respect to the parameter count n to be n^-1/3, as shown above. This means a network with twice the size only gives an error reduction of ca. 20% … that’s not a lot.

Interestingly, the measured convergence rates are agnostic to the physical system and the studied network architectures. For prediction setups (pure NN, no solver), the convergence rate of the networks with respect to the parameter count n is even slightly worse with n^-1/3 , as shown below.

Interestingly, even for larger “foundation-model” networks there seems to be a similar scaling for the accuracy over parameters: here’s a comparison with the Poseidon paper (very interesting: which uses a transformer architecture , side-by-side with our experiments with a simpler ConvNet (same as above). We saw n^-1/4 , there it seems closer to n^-1/6:

Conclusions: This convergence rate is poor compared to classic numerical solvers, and indicates that neural networks are best applied for their intrinsic benefits. They possess appealing characteristics like data-driven fitting, reduced modeling biases, and flexible applications. In contrast, scaling to larger problems is more efficiently achieved by numerical approaches. In applications, it is thus advisable to combine both methods to render the benefits of both components. It also motivates the correction hybrids, where a NN supports a numerical solver. These achieve much higher accuracies, the solver can take care of the large scale generalization, and the NN can be correspondingly smaller.

For details, please check out the full paper at:

Talk about Diffusion Models for Probabilistic Neural Solvers

Here’s also a talk summarizing our recent work on diffusion models for probabilistic Neural solvers:

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:

How to accurately evaluate whether a diffusion model learned the right distribution?

This is a tough task, as many existing data sets for generative models come without quantifiable ground truth data. In contrast the 1D airfoil case of our recent AIAA paper is highly non-trivial but comes with plenty of GT data. Thus, it’s easy to check whether a neural network such as a diffusion model learned the correct distribution of the solutions (by computing the “coverage” in terms of distance to the GT solutions), and to check how much training data is needed to actually converge.

Here’s a Jupyter notebook that explains how to use it:

The image below shows the content of the data set: the input is a single parameter, the Reynolds number, and with increasing Re the complexity of the solutions rises, and starts to vary more and more. The mean on the left hand side stays largely the same, while the increasing and changing standard deviation of the solutions (show on the right) highlights the enlarged complexity of the solutions. Intuitively, the low Re cases have flows that mostly stick to the mean behavior, while the more turbulent ones have a larger number of different structures from more and more complex vortex shedding. As a consequence, a probabilistic neural network trained on this case will need to figure out how the solutions change along Re. Also, it will need to figure out how to generate the different modes of the solutions that arise for larger Re cases.

For details and comparisons with other approaches, please check out section (B) of the paper. The data of the 1D case itself is checked in at

Neural solvers for wake flows at Physical Review E

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 , and source code is available at

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.

Deep learning-based predictive modeling of transonic flow over aerofoils

Our work on transonic flows over aerofoils is finally online:

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.

Learning Symmetric Basis Convolutions at ICLR’24

Our ICLR’24 paper on learning Fourier-based convolutions (SFBC) for particle and unstructured data is online now on arXiv:

A first version of the SFBC source code is also up at , the approach is especially interesting as an inductive bias for accurate neural networks, e.g. to replace graph-nets.

The graph above shows a quantitative evaluation of different network architectures for a fixed layout with four message-passing steps and 32 features per layer. It’s noticeable that the Fourier-convolutions (SFBC) clearly outperform the graph-net based methods (MLPCConv, GNS and MP-PDE on the right). We noticed this is many settings: for a given parameter budget, the inductive bias of the convolutions helps the network to correlate spatial features, and to give more accurate results.

Visual Flow Capture with Raspberry Pi

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:

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

How Temporal Unrolling Supports Neural Physics Simulators on arXiv

It’s worth pointing out that our paper on “How Temporal Unrolling Supports Neural Physics Simulators” is online now:

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.

Differentiable simulation library Phi-ML at JOSS

The differentiable simulation library ΦML (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:

Short summary: ΦML 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. ΦML is compatible with JaxPyTorchTensorFlow and NumPy and your code can be executed on all of these backends.

Two accepted ICLR 2024 papers: particle simulations & stabilized BPTT

We’re happy to report two accepted papers at ICLR 2024! Congrats Patrick and Rene 😀 👍 They’re on particle-based learning and stabilized backprop through time, 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.

Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models

Our paper & source code on using diffusion models to infer RANS solutions for flows around airfoils is online now. It shows that diffusion models finally provide a reliable way to learn full distributions of solutions!


Here’s an example result, shown in terms of the standard deviation over 100 samples given one set of initial free stream conditions and a fixed airfoil shape:

Footnote: the heteroscedastic version (in blue) is not a competitor, it learns mean and standard deviation well, but can’t produce samples.

Here’s the full paper abstract for completeness: Leveraging neural networks as surrogate models for turbulence simulation is a topic of growing interest. At the same time, embodying the inherent uncertainty of simulations in the predictions of surrogate models remains very challenging. The present study makes a first attempt to use denoising diffusion probabilistic models (DDPMs) to train an uncertainty-aware surrogate model for turbulence simulations. Due to its prevalence, the simulation of flows around airfoils with various shapes, Reynolds numbers, and angles of attack is chosen as the learning objective. Our results show that DDPMs can successfully capture the whole distribution of solutions and, as a consequence, accurately estimate the uncertainty of the simulations. The performance of DDPMs is also compared with varying baselines in the form of Bayesian neural networks and heteroscedastic models. Experiments demonstrate that DDPMs outperform the other methods regarding a variety of accuracy metrics. Besides, it offers the advantage of providing access to the complete distributions of uncertainties rather than providing a set of parameters. As such, it can yield realistic and detailed samples from the distribution of solutions.

Diffusion models and score matching via differentiable physics: paper and source code

The final version of our NeurIPS paper merging physics simulations into the diffusion modeling process (SMDP) is on arXiv now:

Maybe even more importantly, the SMDP source code is online how at: , let us know how it works for you!

Here’s an overview of the algorithm:

Here’s a preview of one of the examples diffusing a very simply decaying “physics” function:

Full paper abstract: Our works proposes a novel approach to solve inverse problems involving the temporal evolution of physics systems by leveraging the idea of score matching. The system’s current state is moved backward in time step by step by combining an approximate inverse physics simulator and a learned correction function. A central insight of our work is that training the learned correction with a single-step loss is equivalent to a score matching objective, while recursively predicting longer parts of the trajectory during training relates to maximum likelihood training of a corresponding probability flow. In the paper, we highlight the advantages of our algorithm compared to standard denoising score matching and implicit score matching, as well as fully learned baselines for a wide range of inverse physics problems. The resulting inverse solver has excellent accuracy and temporal stability and, in contrast to other learned inverse solvers, allows for sampling the posterior of the solutions.

Turbulent Flow Simulation using Autoregressive Conditional Diffusion Models in Action

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:

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:

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:

Learning via Differentiable Physics for Plasma Turbulence

We’re also happy to announce the preprint of our paper on “Physics-Preserving AI-Accelerated Simulations of Plasma Turbulence”: , it’s great to see that training with a differentiable physics solver also yields accurate drift-wave turbulence!

The corresponding source code of Robin Greif’s implementation is also online at: , it contains a fully differentiable Hasagawa-Wakatani solver implemented with PhiFlow. (And a lot of tools for evaluation on top! )

Full paper abstract: Turbulence in fluids, gases, and plasmas remains an open problem of both practical and fundamental importance. Its irreducible complexity usually cannot be tackled computationally in a brute-force style. Here, we combine Large Eddy Simulation (LES) techniques with Machine Learning (ML) to retain only the largest dynamics explicitly, while small-scale dynamics are described by an ML-based sub-grid-scale model. Applying this novel approach to self-driven plasma turbulence allows us to remove large parts of the inertial range, reducing the computational effort by about three orders of magnitude, while retaining the statistical physical properties of the turbulent system.

Diffusion Models for Temporal Predictions of PDEs

Here’s another interesting result from the diffusion-based temporal predictions with ACDM: the diffusion training inherently works with losses computed on single timesteps, but is as stable as a model trained with many steps of unrolling; 16 are needed here:

We were also glad to find out that the diffusion sampling in the strongly conditioned-regime of temporal forecasting works very well with few steps. Instead of 1000 (or so), 50 steps and less already work very well:

The corresponding project page is this one, and the source code can be found at:

Control of Two-way Coupled Fluid Systems with Differentiable Solvers

We finally also have the source code for our RB-control paper online: , the paper being available here:

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.

ACDM Source Code on Github

The source code for our turbulent flow simulations using Autoregressive Conditional Diffusion Models (ACDMs) is online now at , 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.

Hybrid Solver for Reactive Flows Source-Code and Paper accepted at DCE

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:

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.

PDE-based Simulations and Turbulent Flows using Autoregressive Conditional Diffusion Models

Our paper on autoregressive diffusion models for improved temporal predictions of complex PDE-based simulations is online now at (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.