Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs), due to complex geometries, interactions between physical variables, and the lack of large amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to the function space. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations and fluid-structure interactions, we found CoDA-NO to outperform existing methods on the few-shot learning task by over $36\%$. The code is available at https://github.com/ashiq24/CoDA-NO.

Plasma Surrogate Modelling using Fourier Neural Operators

Predicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion. Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors. Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models. We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier Neural Operators (FNO). We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (MSE $\approx$ $10^{-5}$). Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations (PDE), and can capture the dependence among the different variables in a single model. FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e., cameras looking across the central solenoid and the divertor in the Tokamak. We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring. We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full duration of the plasma shot within MAST. The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions.

FusionU-Net: U-Net with Enhanced Skip Connection for Pathology Image Segmentation

In recent years, U-Net and its variants have been widely used in pathology image segmentation tasks. One of the key designs of U-Net is the use of skip connections between the encoder and decoder, which helps to recover detailed information after upsampling. While most variations of U-Net adopt the original skip connection design, there is semantic gap between the encoder and decoder that can negatively impact model performance. Therefore, it is important to reduce this semantic gap before conducting skip connection. To address this issue, we propose a new segmentation network called FusionU-Net, which is based on U-Net structure and incorporates a fusion module to exchange information between different skip connections to reduce semantic gaps. Unlike the other fusion modules in existing networks, ours is based on a two-round fusion design that fully considers the local relevance between adjacent encoder layer outputs and the need for bi-directional information exchange across multiple layers. We conducted extensive experiments on multiple pathology image datasets to evaluate our model and found that FusionU-Net achieves better performance compared to other competing methods. We argue our fusion module is more effective than the designs of existing networks, and it could be easily embedded into other networks to further enhance the model performance.

Neural Operators for Accelerating Scientific Simulations and Design

Scientific discovery and engineering design are currently limited by the time and cost of physical experiments, selected mostly through trial-and-error and intuition that require deep domain expertise. Numerical simulations present an alternative to physical experiments, but are usually infeasible for complex real-world domains due to the computational requirements of existing numerical methods. Artificial intelligence (AI) presents a potential paradigm shift through the development of fast data-driven surrogate models. In particular, an AI framework, known as neural operators, presents a principled framework for learning mappings between functions defined on continuous domains, e.g., spatiotemporal processes and partial differential equations (PDE). They can extrapolate and predict solutions at new locations unseen during training, i.e., perform zero-shot super-resolution. Neural operators can augment or even replace existing simulators in many applications, such as computational fluid dynamics, weather forecasting, and material modeling, while being 4-5 orders of magnitude faster. Further, neural operators can be integrated with physics and other domain constraints enforced at finer resolutions to obtain high-fidelity solutions and good generalization. Since neural operators are differentiable, they can directly optimize parameters for inverse design and other inverse problems. We believe that neural operators present a transformative approach to simulation and design, enabling rapid research and development.

Geometry-Informed Neural Operator for Large-Scale 3D PDEs

We propose the geometry-informed neural operator (GINO), a highly efficient approach to learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function and point-cloud representations of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. GINO is discretization-convergent, meaning the trained model can be applied to arbitrary discretization of the continuous domain and it converges to the continuum operator as the discretization is refined. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a $26,000 \times$ speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

The Nonlocal Neural Operator: Universal Approximation

Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate traditional numerical methods and to enable data-driven discovery. A popular variant of neural operators is the Fourier neural operator (FNO). Previous analysis proving universal operator approximation theorems for FNOs resorts to use of an unbounded number of Fourier modes and limits the basic form of the method to problems with periodic geometry. Prior work relies on intuition from traditional numerical methods, and interprets the FNO as a nonstandard and highly nonlinear spectral method. The present work challenges this point of view in two ways: (i) the work introduces a new broad class of operator approximators, termed nonlocal neural operators (NNOs), which allow for operator approximation between functions defined on arbitrary geometries, and includes the FNO as a special case; and (ii) analysis of the NNOs shows that, provided this architecture includes computation of a spatial average (corresponding to retaining only a single Fourier mode in the special case of the FNO) it benefits from universal approximation. It is demonstrated that this theoretical result unifies the analysis of a wide range of neural operator architectures. Furthermore, it sheds new light on the role of nonlocality, and its interaction with nonlinearity, thereby paving the way for a more systematic exploration of nonlocality, both through the development of new operator learning architectures and the analysis of existing and new architectures.

Forecasting subcritical cylinder wakes with Fourier Neural Operators

We apply Fourier neural operators (FNOs), a state-of-the-art operator learning technique, to forecast the temporal evolution of experimentally measured velocity fields. FNOs are a recently developed machine learning method capable of approximating solution operators to systems of partial differential equations through data alone. The learned FNO solution operator can be evaluated in milliseconds, potentially enabling faster-than-real-time modeling for predictive flow control in physical systems. Here we use FNOs to predict how physical fluid flows evolve in time, training with particle image velocimetry measurements depicting cylinder wakes in the subcritical vortex shedding regime. We train separate FNOs at Reynolds numbers ranging from Re = 240 to Re = 3060 and study how increasingly turbulent flow phenomena impact prediction accuracy. We focus here on a short prediction horizon of ten non-dimensionalized time-steps, as would be relevant for problems of predictive flow control. We find that FNOs are capable of accurately predicting the evolution of experimental velocity fields throughout the range of Reynolds numbers tested (L2 norm error < 0.1) despite being provided with limited and imperfect flow observations. Given these results, we conclude that this method holds significant potential for real-time predictive flow control of physical systems.

Incremental Fourier Neural Operator

Recently, neural networks have proven their impressive ability to solve partial differential equations (PDEs). Among them, Fourier neural operator (FNO) has shown success in learning solution operators for highly non-linear problems such as turbulence flow. FNO is discretization-invariant, where it can be trained on low-resolution data and generalizes to problems with high-resolution. This property is related to the low-pass filters in FNO, where only a limited number of frequency modes are selected to propagate information. However, it is still a challenge to select an appropriate number of frequency modes and training resolution for different PDEs. Too few frequency modes and low-resolution data hurt generalization, while too many frequency modes and high-resolution data are computationally expensive and lead to over-fitting. To this end, we propose Incremental Fourier Neural Operator (IFNO), which augments both the frequency modes and data resolution incrementally during training. We show that IFNO achieves better generalization (around 15% reduction on testing L2 loss) while reducing the computational cost by 35%, compared to the standard FNO. In addition, we observe that IFNO follows the behavior of implicit regularization in FNO, which explains its excellent generalization ability.

Fourier Continuation for Exact Derivative Computation in Physics-Informed Neural Operators

The physics-informed neural operator (PINO) is a machine learning architecture that has shown promising empirical results for learning partial differential equations. PINO uses the Fourier neural operator (FNO) architecture to overcome the optimization challenges often faced by physics-informed neural networks. Since the convolution operator in PINO uses the Fourier series representation, its gradient can be computed exactly on the Fourier space. While Fourier series cannot represent nonperiodic functions, PINO and FNO still have the expressivity to learn nonperiodic problems with Fourier extension via padding. However, computing the Fourier extension in the physics-informed optimization requires solving an ill-conditioned system, resulting in inaccurate derivatives which prevent effective optimization. In this work, we present an architecture that leverages Fourier continuation (FC) to apply the exact gradient method to PINO for nonperiodic problems. This paper investigates three different ways that FC can be incorporated into PINO by testing their performance on a 1D blowup problem. Experiments show that FC-PINO outperforms padded PINO, improving equation loss by several orders of magnitude, and it can accurately capture the third order derivatives of nonsmooth solution functions.

Machine Learning Accelerated PDE Backstepping Observers

State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.