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Clifford Neural Layers for PDE Modeling

Johannes Brandstetter, Rianne van den Berg, Max Welling, Jayesh K. Gupta

arXiv:2209.04934v2cs.LGcs.CVphysics.flu-dyn

TL;DR

Neural PDE surrogates can accelerate simulations, but conventional approaches do not explicitly model relationships among correlated fields and their components. This paper introduces Clifford neural layers that process such fields as multivectors, replacing convolution and Fourier operations with Clifford counterparts. The tested Clifford layers improve generalization over standard neural PDE surrogates for similar parameter counts, while Clifford Fourier layers have a slower backward pass in the reported implementation.

  • Problem

    Neural PDE surrogates do not explicitly model relationships among correlated fields and their components, despite the computational cost of standard PDE solution methods.

    Source: §1 · pp. 1–2

  • Method

    The paper replaces convolution and Fourier operations in neural PDE surrogates with Clifford counterparts that operate on multivector feature fields.

    Source: §1 · p. 2, §3 · pp. 5–6

  • Results

    Clifford neural layers significantly outperform equivalent standard neural PDE surrogates, and tested Clifford counterparts improve results across Navier–Stokes, weather-modeling, and Maxwell tasks.

    Source: §4 · pp. 8–9, §5 · p. 9

  • Takeaways & Limitations

    The multivector viewpoint represents relationships among fields and their components as geometric objects, providing an inductive bias advantage in the tested neural PDE surrogates.

    Source: §5 · p. 9

    Clifford Fourier layers use complex-valued FFT operations whose backward pass is approximately twice as slow as for FNO layers; FNO and CFNO inference times are similar for similar parameter counts.

    Source: §5 · p. 9

Abstract

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Partial differential equations (PDEs) see widespread use in sciences and engineering to describe simulation of physical processes as scalar and vector fields interacting and coevolving over time. Due to the computationally expensive nature of their standard solution methods, neural PDE surrogates have become an active research topic to accelerate these simulations. However, current methods do not explicitly take into account the relationship between different fields and their internal components, which are often correlated. Viewing the time evolution of such correlated fields through the lens of multivector fields allows us to overcome these limitations. Multivector fields consist of scalar, vector, as well as higher-order components, such as bivectors and trivectors. Their algebraic properties, such as multiplication, addition and other arithmetic operations can be described by Clifford algebras. To our knowledge, this paper presents the first usage of such multivector representations together with Clifford convolutions and Clifford Fourier transforms in the context of deep learning. The resulting Clifford neural layers are universally applicable and will find direct use in the areas of fluid dynamics, weather forecasting, and the modeling of physical systems in general. We empirically evaluate the benefit of Clifford neural layers by replacing convolution and Fourier operations in common neural PDE surrogates by their Clifford counterparts on 2D Navier-Stokes and weather modeling tasks, as well as 3D Maxwell equations. For similar parameter count, Clifford neural layers consistently improve generalization capabilities of the tested neural PDE surrogates. Source code for our PyTorch implementation is available at https://microsoft.github.io/cliffordlayers/.

1 INTRODUCTION

Neural PDE surrogates address the high computational cost of numerical solutions, but typical approaches omit geometric relationships among field components. The paper represents related fields as multivectors and replaces neural-network operations with Clifford-algebra counterparts.

1 INTRODUCTION

  • Numerical solutions to many physical equations are analytically intractable and can require computationally prohibitive approximation schemes.The introduction identifies fluid mechanics and electromagnetism as examples, including Navier–Stokes and Maxwell equations.Source: §1 · p. 1
  • Current deep-learning approaches stack scalar fields and vector components alike, omitting geometric relationships within and between fields.The introduction notes that such relationships can provide inductive-bias information, using wind components and pressure as examples.Source: §1 · p. 2
  • The paper replaces feature-field operations with Clifford-algebra operations on multivector fields that include scalar, vector, and higher-order components.The proposed convolution kernels have multivector components and convolve over multivector feature maps.Source: §1 · p. 2

2 BACKGROUND: CLIFFORD ALGEBRAS

Clifford algebras provide a geometric-algebra framework whose multivectors combine scalar, vector, and higher-grade components. The section introduces algebraic operations and examples that support the neural layers developed later.

2 BACKGROUND: CLIFFORD ALGEBRAS

  • A Clifford algebra Clp,q(R) is an associative algebra generated by orthonormal basis elements whose quadratic relations depend on its signature (p, q).Its vector space includes elements representing plane and volume segments as well as scalars and vectors.Source: §2 · p. 3
  • Multivectors combine elements of grade-specific subspaces, including scalars, vectors, bivectors, and higher-grade elements.The geometric product is bilinear and associative but is generally non-commutative; the dual relates different multivectors using the pseudoscalar.Source: §2 · p. 3
  • The wedge product of two vectors is antisymmetric, x ∧ y = −y ∧ x, and represents an oriented plane segment.The unit bivector in two dimensions squares to −1 and acts as a geometric analogue of √−1.Source: §2 · p. 4
  • Cl0,2(R) is isomorphic to the quaternions, while Cl3,0(R) includes scalar, vector, bivector, and trivector components.In the paper’s electromagnetic example, the field combines electric vector components with magnetic bivector components, illustrating how multivectors represent both kinds of field.Source: §2 · pp. 3–4

3 CLIFFORD NEURAL LAYERS

The paper constructs Clifford counterparts to convolutional and Fourier neural layers by operating on multivector feature fields. It also describes a rotational convolution variant and multivector Fourier processing in two and three dimensions.

3 CLIFFORD NEURAL LAYERS

  • Clifford convolutions replace scalar products between feature values and filters with geometric products of multivector inputs and kernels.Each product yields a multivector, so the layer output is a grid of multivectors; the 2D construction uses four kernels for one multivector kernel.Source: §3 · p. 5
  • The rotational Clifford CNN layer uses the Cl0,2(R)-quaternion isomorphism to parameterize multivector transformations as rotations.Its scalar response is combined with rotations acting on vector and bivector parts; the paper reports better experimental performance than the equally flexible Clifford CNN layer.Source: §3 · p. 5
  • Clifford convolutions and rotational Clifford CNN layers satisfy translation equivariance for multivector inputs.The paper points to an appendix theorem for the convolution result and says the rotational-layer property follows analogously.Source: §3 · p. 6
  • The 2D Clifford Fourier transform combines two standard Fourier transforms applied to dual-pair components represented as complex-valued signals.The resulting multivector Fourier modes are modified pointwise by geometric products with multivector weights, and the residual connection is a Clifford convolution.Source: §3 · p. 6
  • For Cl3,0(R), the Fourier construction applies four transforms to paired scalar/trivector and vector/bivector components.Each pair is combined into a complex field before Fourier transformation.Source: §3 · p. 6

4 EXPERIMENTS

The experiments compare baseline and Clifford counterparts on Navier–Stokes, shallow-water weather modeling, and 3D Maxwell tasks. Performance is assessed with one-step and rollout losses.

4 EXPERIMENTS

  • The study replaces baseline convolutions and Fourier transforms with Clifford counterparts while keeping parameter counts similar across the tested PDE surrogates.The tasks include one scalar and one 2D vector field for Navier–Stokes and shallow water, and a 3D electric vector field with its dual magnetic bivector field for Maxwell.Source: §4 · pp. 6–7
  • One-step loss measures next-timestep mean-squared error summed over fields, while rollout loss measures error after five surrogate applications, summed over fields and time.Navier–Stokes inputs comprise a two-timestep history.Source: §4 · p. 7, Figure 6 · p. 7
  • The Navier–Stokes setup includes a transported scalar field alongside velocity, with the scalar influencing velocity only through an external force term.The data are generated on a 128 × 128 spatial grid.Source: §4 · p. 7
  • CFNO architectures improve on FNO architectures across dataset sizes for both one-step and rollout losses in the Navier–Stokes experiments.The figure presents results for ResNet-based and Fourier-based architectures.Source: §4 · p. 8, Figure 7 · p. 8
  • For shallow-water weather modeling, baseline–Clifford performance differences are more pronounced, particularly where the scalar and vector fields are strongly coupled.The shallow-water equations model a thin fluid layer through three coupled PDEs describing two velocity components and scalar pressure.Source: §4 · p. 8
  • On 3D Maxwell tasks, CFNO improves on FNO, especially with fewer trajectories, supporting the use of Clifford Fourier layers for 3D problems.The test fields are the vector-valued D field and H field, evaluated on a 32 × 32 × 32 grid.Source: §4 · p. 9

5 CONCLUSION

The conclusion presents Clifford neural layers as geometric representations of scalar, vector, bivector, and higher-order fields, and reports that they significantly outperform equivalent standard neural PDE surrogates. It also notes training-speed and retraining constraints, outlines future applications, and states reproducibility and ethical considerations.

5 CONCLUSION

  • Clifford layers organize scalar, vector, bivector, and higher-order fields as multivectors and significantly outperform equivalent standard neural PDE surrogates.The authors attribute this advantage to better representation of relationships between fields and their components.Source: §5 · p. 9
  • Clifford convolutions increase training time by about 2×, while PDE surrogates must be retrained for different equation parameters or time steps.For similar parameter counts, FNO and CFNO inference times are similar; Clifford convolutions are more parameter-efficient but use more operations.Source: §5 · p. 9
  • The authors identify MRI, radar, neural implicit representations, graph networks, attention, custom multivector GPU kernels, Julia, and physics-informed methods as future directions.They suggest custom multivector GPU kernels could address speed issues.Source: §5 · p. 9

REPRODUCIBILITY AND ETHICAL STATEMENT

  • The PyTorch implementation is open-sourced, with architecture descriptions and layer pseudocode provided for reproducibility.The authors also report including error bars and ablation studies where considered necessary and appropriate.Source: REPRODUCIBILITY AND ETHICAL STATEMENT · p. 10
  • The authors note that PDE surrogates may offer alternatives to computationally expensive simulations, while emphasizing rigorous cross-checks and monitoring.They describe potential relevance to fluid dynamics and weather modeling and a possible connection to reducing the carbon footprint.Source: REPRODUCIBILITY AND ETHICAL STATEMENT · p. 10

Appendices

The appendices develop the mathematical background for Clifford algebras and multivector representations, relating them to electromagnetism and Clifford neural layers. They detail Clifford convolution and Fourier-layer constructions, their implementation adaptations and equivariance properties, and a rotational quaternion parameterization.

A.1 CLIFFORD ALGEBRAS

  • Clifford algebras organize scalars, vectors, and higher-grade elements as multivectors, with a bilinear geometric product that combines them.The algebra generated from R^n has dimension 2^n.Source: §A.1 · pp. 22–23

A.2.1 CLIFFORD ALGEBRA Cl0,1(R)

  • Cl0,1(R) is a two-dimensional algebra isomorphic to the complex numbers, with e1 serving as the imaginary unit.Its basis is {1, e1}, and e1 squares to −1.Source: §A.2.1 · p. 24

A.2.2 CLIFFORD ALGEBRA Cl2,0(R)

  • In Cl2,0(R), the bivector e1e2 represents the plane spanned by e1 and e2, squares to −1, and supports dual pairs between scalar and bivector, and between vectors.The algebra’s four basis elements are {1, e1, e2, e1e2}.Source: §A.2.2 · pp. 24–25

A.2.3 CLIFFORD ALGEBRA Cl0,2(R)

  • Cl0,2(R) is a four-dimensional Clifford algebra isomorphic to the quaternions, whose three imaginary basis elements square to −1.Its basis is {1, e1, e2, e1e2}.Source: §A.2.3 · p. 25

A.2.4 CLIFFORD ALGEBRA Cl3,0(R)

  • Cl3,0(R) has eight basis elements: one scalar, three vectors, three bivectors, and a trivector pseudoscalar.Its multivector product is defined analogously to that of Cl2,0(R).Source: §A.2.4 · p. 26

A.3 THE ELECTROMAGNETIC FIELD IN 3 DIMENSIONS

  • In three dimensions, the electromagnetic field combines electric vectors with magnetic bivectors, representing the magnetic pseudovector through its dual.The appendix notes that electric and magnetic fields are strongly coupled, and that the multivector field comprises three vectors and three bivectors.Source: §A.3 · p. 27

B CLIFFORD NEURAL LAYERS

B.1 CLIFFORD CONVOLUTION LAYERS

  • The appendix derives translation-equivariant Clifford convolution layers for multivectors in G2 and extends them to algebras generated by R3.It introduces regular CNN layers as a point of comparison.Source: §B.1 · p. 28

B.1 CLIFFORD CONVOLUTION LAYERS; B.1.1 TRANSLATION EQUIVARIANCE OF CLIFFORD CONVOLUTIONS

  • Clifford convolutions replace scalar feature maps and elementwise products with multivector maps and geometric products, and are translation equivariant.The translation property is illustrated by shifting the input and kernel arguments together.Source: §B.1 · p. 28, §B.1.1 · p. 29, §B.4 · p. 31

B.1.2 ROTATIONAL CLIFFORD CNN LAYERS

  • The Cl0,2(R)-to-quaternion isomorphism yields a rotational filter parameterization combining a scalar response with quaternion rotation of vector components.The rotation acts on the vector part of the feature map and is built from the filter’s scalar and vector parts.Source: §B.1.2 · pp. 29–30

B.1.3 3D CLIFFORD CONVOLUTION LAYERS

  • A 3D Cl3,0(R) convolution uses eight convolution kernels for one multivector kernel, covering scalar, vector, bivector, and trivector input parts.Other 3D Clifford algebras alter the geometric-product signs.Source: §B.1.3 · p. 30

B.2 CLIFFORD NORMALIZATION

  • Multivector normalization is formulated as whitening 4D vectors because separately centering and scaling components does not ensure equal variance across them.The learnable shift has four components and the scaling parameter is a positive 4 × 4 matrix.Source: §B.2 · pp. 30–31

B.3 CLIFFORD INITIALIZATION

  • Quaternion initialization schemes adapt to Clifford layers, but showed no gains over default PyTorch initialization in 2D experiments; 3D implementations require initialization values reduced by a factor of 1/8.Source: §B.3 · p. 31

B.4 EQUIVARIANCE UNDER ROTATIONS AND REFLECTIONS

  • Clifford convolutions are translation equivariant but not inherently equivariant to multivector rotations or reflections; a kernel constraint enables the latter equivariance.When the input transforms as a multivector and the kernel satisfies the constraint, the output also transforms properly.Source: §B.4 · p. 31

B.5 CLIFFORD FOURIER LAYERS; B.5.1 2D CLIFFORD FOURIER TRANSFORM

  • The appendices derive Clifford Fourier layers for G2 and G3; in 2D, the transform combines classical transforms of spinor and vector complex pairs with pointwise multivector weights and mode cutoffs.As in Fourier neural operators, modes above cutoff frequencies are zeroed, and the residual connection is replaced by a Clifford convolution.Source: §B.5 · p. 32, §B.5.1 · pp. 33–34

B.5.2 2D CLIFFORD CONVOLUTION THEOREM

The appendix establishes a 2D Clifford convolution theorem for right-applied multivector filters by distinguishing commuting and anti-commuting parts. It then constructs the 3D Clifford Fourier transform from four classical transforms and describes its use in Fourier layers.

B.5.2 2D CLIFFORD CONVOLUTION THEOREM

  • The 2D convolution theorem treats the spinor part as commuting with the Fourier kernel and the vector part as anti-commuting, for Cl(2, 0)(R) and Cl(0, 2)(R).The proof is formulated for multivector-valued filters applied from the right, consistent with the Clifford convolution layers.Source: §B.5.2 · p. 34

B.5.3 3D CLIFFORD FOURIER TRANSFORM

  • The 3D Clifford Fourier transform combines four standard transforms by pairing scalar and trivector components and pairing each vector component with its dual bivector.Each pair is treated as a complex-valued signal; together they form an element of C^4.Source: §B.5.3 · p. 35
  • The 3D Fourier layer applies multivector weight tensors pointwise, zeros modes above the cutoff frequency, and replaces FNO residual weights with Clifford convolution.The Fourier modes follow from combining the four parts of the transform.Source: §B.5.3 · p. 36

B.5.4 3D CLIFFORD CONVOLUTION THEOREM

The 3D convolution theorem checks how the Fourier kernel interacts with multivector parts. It establishes that all four parts commute with the pseudoscalar across 3D Clifford algebras.

B.5.4 3D CLIFFORD CONVOLUTION THEOREM

  • The proof checks whether the Fourier-transform basis elements commute with the pseudoscalar i3.It examines the scalar and three vector basis elements.Source: §B.5.4 · p. 36
  • The scalar and vector basis elements commute with i3 in the displayed 3D relations.These relations provide the commutation check used in the theorem argument.Source: §B.5.4 · p. 36
  • All four multivector parts commute with i3, a property stated to hold for all 3D Clifford algebras.The appendix contrasts this with the 2D Clifford Fourier transform.Source: §B.5.4 · p. 36

B.5.5 IMPLEMENTATION OF CLIFFORD FOURIER LAYERS

Clifford Fourier layers transform multivector components through dual pairs, multiply Fourier modes with the geometric product, and reconstruct the spatial-domain multivector. The pseudocode also specifies ordinary and rotational Clifford convolutions and combines spectral and convolution paths in a 2D layer.

B.5.5 IMPLEMENTATION OF CLIFFORD FOURIER LAYERS

  • 2D Clifford Fourier layers apply two standard Fourier transforms to dual pairs, multiply modes with the geometric product, then inverse-transform and reassemble the multivector.The 3D implementation applies four standard Fourier transforms to dual pairs; the convolution theorems apply to the vector and spinor parts and the dual pairs in Cl2,0 and Cl3,0.Source: §B.5.5 · p. 37

B.6 PSEUDOCODE

  • The appendix pseudocode presents algorithms for Clifford convolution, rotational Clifford convolution, and a Clifford Fourier layer.The provided 2D Fourier-layer algorithm uses Cl2,0.Source: §B.6 · pp. 37–38
  • A 2D Clifford convolution views multivector inputs as real vectors, applies a convolution using a Clifford kernel, and views the output as a multivector.The pseudocode also supplies the kernel component arrangement for Cl2,0.Source: §B.6 · p. 37
  • The rotational 2D Clifford convolution constructs its kernel from normalized pairwise products of weights before applying ordinary convolution to a real-vector view of the input.Its pseudocode labels the operation as using Cl0,2.Source: §B.6 · p. 38
  • The 2D Clifford Fourier layer adds a spectral Clifford convolution to a real-vector Clifford convolution, then applies an activation.The pseudocode forms multivector Fourier modes from scalar and vector components, applies weights using the geometric product, and inverse-transforms the vector and scalar parts.Source: §B.6 · p. 38

C EXPERIMENTS

This appendix supports Section 4 of the main paper.

C EXPERIMENTS

  • The appendix supports Section 4 of the main paper.The supplied passage states its relationship to the main paper but gives no additional experimental details.Source: C EXPERIMENTS · p. 39

C.1 LOSS FUNCTION AND METRICS

The experiments report summed MSE and define one-step, vector, scalar, and rollout losses by selecting the prediction horizon and included fields. Maxwell electric and magnetic losses are defined analogously to selected Navier–Stokes and shallow-water losses.

C.1 LOSS FUNCTION AND METRICS

  • The reported loss is summed MSE (SMSE).The supplied text introduces its definition but does not include the formula.Source: §C.1 · p. 39
  • The one-step loss uses Nt = 1 and includes all scalar and vector components, while rollout loss uses Nt = 5 and includes both.Equation 79 is used for training with Nt = 1; its definition identifies u as the target, model output as û, Nfields as selected scalar fields and vector components, and Ny as the number of spatial points.Source: §C.1 · p. 39
  • Vector loss uses only vector components, while scalar loss uses only the scalar field; both use Nt = 1.Source: §C.1 · p. 39
  • Maxwell electric and magnetic losses are defined analogously to Navier–Stokes and shallow-water vector and scalar losses, respectively.Source: §C.1 · p. 39

C.2 MODELS

The experiments compare fine-tuned ResNet and Fourier Neural Operator architectures with Clifford counterparts at similar parameter counts. The appendix specifies configurations for 2D Navier–Stokes and shallow-water experiments and 3D Maxwell experiments.

C.2 MODELS

  • Baseline models from two architecture families, ResNets and FNOs, are fine-tuned per experiment; Clifford versions replace layers, normalization, and nonlinearities with their Clifford counterparts.The best models are reported, and Clifford non-linearities apply the corresponding default linearity to different multivector components.Source: §C.2 · p. 39
  • The 2D Navier–Stokes and shallow-water ResNet baseline uses eight residual blocks, 128 hidden channels, constant spatial resolution, and roughly 2.4 million parameters.Its blocks use two 3×3 convolution layers, shortcut connections, group normalization, and GeLU; more blocks or channels did not significantly improve performance.Source: §C.2 · p. 39
  • Clifford ResNets replace CNN layers with Clifford or rotational Clifford CNN layers and use 64 multivector channels, yielding roughly 1.6 million parameters.They also use Clifford GeLU and Clifford group normalization; Clifford initialization did not improve results.Source: §C.2 · p. 39
  • The 2D FNO baseline uses eight blocks, 128 hidden channels, 16 Fourier modes, and roughly 140 million parameters; Clifford FNOs use 48 multivector channels at roughly the same parameter count.Additional shortcuts, normalization, larger channel counts, or more blocks did not improve baseline performance; Clifford initialization did not improve results.Source: §C.2 · pp. 39–40
  • The 3D Maxwell FNO baseline uses four blocks, 96 hidden channels, six Fourier modes, and roughly 65 million parameters; its Clifford counterpart uses 32 multivector channels and roughly 55 million.More baseline layers or modes degraded performance; Clifford initialization was important because overly large Clifford-convolution weight initial values hindered gradient flow through Clifford Fourier operations.Source: §C.2 · p. 40

C.3 TRAINING AND MODEL SELECTION.

Baseline ResNet and FNO architectures were tuned across layer, channel, normalization, activation, and mode choices; Clifford counterparts replaced convolution and Fourier layers, with fewer layers used to match parameter counts.

C.3 TRAINING AND MODEL SELECTION.

  • Models were trained for 50 epochs with Adam, candidate learning rates [10−4, 2· 10−4, 5 · 10−4], cosine annealing, and linear warmup while minimizing summed mean squared error.Baseline ResNet and FNO architectures were tuned over layers, channels, normalization, activation functions, and Fourier modes; larger layer or channel counts did not improve performance.Source: §C.3 · p. 40
  • Clifford counterparts replaced convolution and Fourier layers, added Clifford normalization schemes, and used fewer layers to obtain similar parameter counts.The authors note that further Clifford-specific architecture optimization was possible but would depart from plug-and-play replacement and add computational overhead.Source: §C.3 · p. 40
  • Clifford runs took about twice as long to train as equivalent architectures and epochs, with average experiment times ranging from 3 h to 48 h.FNO and CFNO experiments used 4×16 GB NVIDIA V100 machines; ResNet and Clifford ResNet experiments used 8×32 GB NVIDIA V100 machines.Source: §C.3 · p. 40

C.4 NAVIER-STOKES IN 2D

The 2D Navier–Stokes experiments model velocity and transported scalar fields, using simulated trajectories for surrogate training. Clifford Fourier models improve on FNO across dataset sizes, while Clifford ResNets improve on ResNet.

C.4 NAVIER-STOKES IN 2D

  • The incompressible Navier–Stokes setup describes velocity flow through momentum conservation and enforces mass conservation with an incompressibility constraint.The modeled scalar field is transported along the velocity field, as in smoke concentration advected by airflow.Source: §C.4 · p. 41
  • The simulations used ΦFlow with pressure-gradient subtraction, semi-Lagrangian advection for velocity, and MacCormack advection for the scalar field.The setup used a 128 × 128 grid, 1.5 s time steps, a 21 s simulation, and trajectories sampled at 14 time points.Source: §C.4 · p. 41
  • CFNO architectures improve upon FNO architectures for all dataset sizes in both one-step and rollout losses.Figure 10 reports rollout, one-step, scalar, and vector losses for FNO and CFNO models trained with increasing numbers of trajectories.Source: §C.4 · p. 41, Figure 10 · p. 43
  • For ResNet-like models, CResNet and CResNetrot improve on ResNet, and their rollout losses are lower; CResNetrot generally performs better than CResNet.The authors attribute lower rollout losses to more stable models that are less prone to overfitting one-step predictions.Source: §C.4 · p. 41
  • In the low-data regime, ResNet-based architectures have much higher loss than FNO-based architectures.Figure 9 reports unrolled, one-step, scalar, and vector losses for ResNet, CResNet, and CResNetrot across training-set sizes.Source: Figure 9 · p. 42

C.5 SHALLOW WATER EQUATIONS.

This section introduces scalar and vector fields for characterizing fluid motion and examines Clifford Fourier and ResNet surrogates on shallow-water simulations. The Clifford surrogates improve on their standard counterparts, with especially pronounced differences attributed to stronger coupling between scalar and vector fields.

C.5 SHALLOW WATER EQUATIONS.

  • The modified SpeedyWeather.jl dataset uses randomized initial conditions and 20-day periodic-domain simulations on a 192 × 96 grid, sampled every 6 hours into 84-time-point pressure and wind trajectories.The implementation combines spherical harmonics for linear terms and a Gaussian grid for nonlinear terms, with spectral transforms and a leapfrog scheme using a Robert and William’s filter for third-order accuracy.Source: §C.5 · p. 45
  • CFNO improves over FNO across dataset sizes for one-step and rollout losses; 56 trajectories yield similar (C)FNO test performance to ResNet models trained on 896.The improvements are especially pronounced with fewer training trajectories.Source: §C.5 · p. 45, Figure 12 · p. 46, Figure 13 · p. 47
  • CResNet and CResNetrot achieve much lower rollout losses than ResNet, and CResNetrot architectures generally perform better than CResNet.The paper attributes the larger-than-Navier–Stokes performance differences between baseline and Clifford architectures to stronger coupling of scalar and vector fields.Source: §C.5 · p. 45, Figure 14 · p. 48
  • Clifford Fourier surrogate rollouts of scalar pressure and vector wind fields are described as fairly indistinguishable from ground truth trajectories.Figure 15 contrasts example surrogate rollouts with ground truth.Source: §C.5 · p. 45, Figure 15 · p. 49

E GLOSSARY

  • Scalar fields assign a scalar to each spatial point, while vector fields assign vectors; examples include temperature, force, electric current, and velocity.The passages define scalar fields as maps D → R and vector fields as maps D → R^n, where D ⊆ R^n.Source: E GLOSSARY · p. 57, Table 6 · p. 58
  • Streamlines are curves tangent to a flow’s velocity vectors, which indicate the direction a massless fluid element travels and characterize the fluid.Streamlines change over time when the fluid flow changes.Source: E GLOSSARY · p. 58

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