Recent advances in mechanistic interpretability have revealed that large language models (LLMs) develop internal representations corresponding not only to concrete entities but also distinct, human-understandable abstract concepts and behaviour. Moreover, these hidden features can be directly manipulated to steer model behaviour. However, it remains an open question whether this phenomenon is unique to models trained on inherently structured data (ie. language, images) or if it is a general property of foundation models. In this work, we investigate the internal representations of a large physics-focused foundation model. Inspired by recent work identifying single directions in activation space for complex behaviours in LLMs, we extract activation vectors from the model during forward passes over simulation datasets for different physical regimes. We then compute "delta" representations between the two regimes. These delta tensors act as concept directions in activation space, encoding specific physical features. By injecting these concept directions back into the model during inference, we can steer its predictions, demonstrating causal control over physical behaviours, such as inducing or removing some particular physical feature from a simulation. These results suggest that scientific foundation models learn generalised representations of physical principles. They do not merely rely on superficial correlations and patterns in the simulations. Our findings open new avenues for understanding and controlling scientific foundation models and has implications for AI-enabled scientific discovery.

This paper investigates the capability of Neural Networks (NNs) as alternatives to the traditional methods to analyse the performance of aerofoils used in the wind and tidal energy industry. The current methods used to assess the characteristic lift and drag coefficients include Computational Fluid Dynamics (CFD), thin aerofoil and panel methods, all face trade-offs between computational speed and the accuracy of the results and as such NNs have been investigated as an alternative with the aim that it would perform both quickly and accurately. As such, this paper provides a benchmark for the windAI_bench dataset published by the National Renewable Energy Laboratory (NREL) in the USA. In order to validate the methodology of the benchmarking, the AirfRANSdataset benchmark is used as both a starting point and a point of comparison. This study evaluates four neural networks (MLP, PointNet, GraphSAGE, GUNet) trained on a range of aerofoils at 25 angles of attack (4$^\circ$ to 20$^\circ$) to predict fluid flow and calculate lift coefficients ($C_L$) via the panel method. GraphSAGE and GUNet performed well during the training phase, but underperformed during testing. Accordingly, this paper has identified PointNet and MLP as the two strongest models tested, however whilst the results from MLP are more commonly correct for predicting the behaviour of the fluid, the results from PointNet provide the more accurate results for calculating $C_L$.

Data assimilation has become a crucial technique aiming to combine physical models with observational data to estimate state variables. Traditional assimilation algorithms often face challenges of high nonlinearity brought by both the physical and observational models. In this work, we propose a novel data-driven assimilation algorithm based on generative models to address such concerns. Our State-Observation Augmented Diffusion (SOAD) model is designed to handle nonlinear physical and observational models more effectively. The marginal posterior associated with SOAD has been derived and then proved to match the real posterior under mild assumptions, which shows theoretical superiority over previous score-based assimilation works. Experimental results also indicate that our SOAD model may offer improved accuracy over existing data-driven methods.

In this paper we introduce a novel Neural Networks-based approach for approximating solutions to the (2D) incompressible Navier--Stokes equations. Our algorithm uses a Physics-informed Neural Network, that approximates the vorticity based on a loss function that uses a computationally efficient formulation of the Random Vortex dynamics. The neural vorticity estimator is then combined with traditional numerical PDE-solvers for the Poisson equation to compute the velocity field. The main advantage of our method compared to standard Physics-informed Neural Networks is that it strictly enforces physical properties, such as incompressibility or boundary conditions, which might otherwise be hard to guarantee with purely Neural Networks-based approaches.

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels.

Researchers have long been numerically solving the partial differential equations that govern important fluid phenomena such as weather, fusion plasmas, and aerodynamics. Of course, the accuracy of the results is always limited by the finite precision and spatial resolution of computer representations of the equations. Computers have also become a powerful tool for exact, rigorous mathematics. Proof assistants, for example, instill confidence a logical argument is sound and all cases have been considered. Programs can tirelessly examine superhumanly large libraries of combinations, such as those underlying the four-color map theorem proof in 1976.

Despite their ubiquity throughout science and engineering, only a handful of partial differential equations (PDEs) have analytical, or closed-form solutions. This motivates a vast amount of classical work on numerical simulation of PDEs and more recently, a whirlwind of research into data-driven techniques leveraging machine learning (ML). A recent line of work indicates that a hybrid of classical numerical techniques and machine learning can offer significant improvements over either approach alone. In this work, we show that the choice of the numerical scheme is crucial when incorporating physics-based priors. We build upon Fourier-based spectral methods, which are known to be more efficient than other numerical schemes for simulating PDEs with smooth and periodic solutions. Specifically, we develop ML-augmented spectral solvers for three common PDEs of fluid dynamics. Our models are more accurate (2-4x) than standard spectral solvers at the same resolution but have longer overall runtimes (~2x), due to the additional runtime cost of the neural network component. We also demonstrate a handful of key design principles for combining machine learning and numerical methods for solving PDEs.

Given sparse observations of buoy velocities, oceanographers are interested in reconstructing ocean currents away from the buoys and identifying divergences in a current vector field. As a first and modular step, we focus on the time-stationary case - for instance, by restricting to short time periods. Since we expect current velocity to be a continuous but highly non-linear function of spatial location, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current reconstruction and divergence identification, due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method with theory and experiments on synthetic and real ocean data.

We present a method to downscale idealized geophysical fluid simulations using generative models based on diffusion maps. By analyzing the Fourier spectra of images drawn from different data distributions, we show how one can chain together two independent conditional diffusion models for use in domain translation. The resulting transformation is a diffusion bridge between a low resolution and a high resolution dataset and allows for new sample generation of high-resolution images given specific low resolution features. The ability to generate new samples allows for the computation of any statistic of interest, without any additional calibration or training. Our unsupervised setup is also designed to downscale images without access to paired training data; this flexibility allows for the combination of multiple source and target domains without additional training. We demonstrate that the method enhances resolution and corrects context-dependent biases in geophysical fluid simulations, including in extreme events. We anticipate that the same method can be used to downscale the output of climate simulations, including temperature and precipitation fields, without needing to train a new model for each application and providing a significant computational cost savings.

Deep reinforcement learning (DRL) for fluidic pinball, three individually rotating cylinders in the uniform flow arranged in an equilaterally triangular configuration, can learn the efficient flow control strategies due to the validity of self-learning and data-driven state estimation for complex fluid dynamic problems. In this work, we present a DRL-based real-time feedback strategy to control the hydrodynamic force on fluidic pinball, i.e., force extremum and tracking, from cylinders' rotation. By adequately designing reward functions and encoding historical observations, and after automatic learning of thousands of iterations, the DRL-based control was shown to make reasonable and valid control decisions in nonparametric control parameter space, which is comparable to and even better than the optimal policy found through lengthy brute-force searching. Subsequently, one of these results was analyzed by a machine learning model that enabled us to shed light on the basis of decision-making and physical mechanisms of the force tracking process. The finding from this work can control hydrodynamic force on the operation of fluidic pinball system and potentially pave the way for exploring efficient active flow control strategies in other complex fluid dynamic problems.