Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders

Neural surrogates enable orders-of-magnitude acceleration of computational fluid dynamics (CFD) simulations, with the potential to transform engineering and healthcare workflows. Neural surrogate use in real-world applications requires addressing scalability to large, high-resolution surface and volume meshes, as well as to bespoke architectures, and accounting for limited training data through the use of inductive biases. Group-equivariant architectures are a principled way to introduce such bias, yet they can be detrimental when the learning problem itself breaks symmetry, for example, due to strong distributional alignment in the dataset. In this work, we investigate under which conditions equivariance improves generalization in neural CFD surrogates across tasks with increasing levels of distributional alignment and realism, covering automotive aerodynamics and blood flow (hemodynamics). To systematically assess the added value of equivariance at the limit of problem scaling, we introduce the Anchored-Branched Geometric Algebra Transformer (AB-GATr), a neural surrogate that integrates scalability and symmetry preservation to efficiently model coupled surface and volume quantities in an $E(3)$-equivariant manner. We find that on strongly aligned aerodynamics datasets, i.e., those that break symmetry, enforcing equivariance can degrade in-distribution performance. In contrast, across hemodynamic benchmarks with diverse geometries and varying alignment, equivariance is consistently beneficial. Moreover, across all benchmarks, the explicit equivariance of AB-GATr reliably outperforms implicit symmetry learning through data augmentation. Our findings showcase that equivariance is not universally beneficial across domains, yet it brings tangible advantages in problems lacking strong data regularities.

We propose an improved Fourier Neural Operator (FNO) for modeling two-dimensional Rayleigh-Bénard convection by predicting time increments instead of full solutions, achieving higher accuracy than a standard FNO baseline. The resulting model is compact (314k parameters, 1.26 MB) and fast (7 ms inference), while maintaining similar accuracy as demonstrated in previous benchmarks. We show that although FNOs generalize to finer meshes, accuracy remains limited by the resolution of the training data.

Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of parameter-dependent high-dimensional dynamical systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose an invertible AE architecture, named inv-AE, that computationally improves upon the stagnation of the reconstruction error typical of traditional AE architectures. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a 1-dimensional Burgers' equation, a 2-dimensional fluid flow around an obstacle with variable geometry, and a 3-dimensional Korteweg-de Vries, we show that (i) inv-AE mitigates the issue of the characteristic plateau of AEs and (ii) inv-AE can be combined with popular autoencoder-based ROM approaches, e.g., DL-ROM, to improve their accuracy.