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REVIEW 3 major objections 6 minor 146 references

Symmetry-preserving neural networks in lattice field theories

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Neural networks that respect gauge symmetry by construction can regress Wilson-loop observables accurately, while symmetry-breaking networks of comparable size fail as the loops grow.

desk verdict Solid compilation of the author's own L-CNN work, with a genuinely useful layer construction but a headline comparison that is confounded by asymmetric supervision and a universality claim that is not fully proven. read the letter →

arxiv 2506.12493 v1 pith:3LQIFG25 submitted 2025-06-14 hep-lat cs.LG

Matteo Favoni This is my paper
classification hep-latcs.LG MSC 68T0781T2581T13 PACS 11.15.Ha07.05.Mh
keywords latticefieldtheorygaugeequivarianceequivariantneuralnetworksWilsonloopsL-CNNtranslationalsymmetrygradientflowfluxrepresentation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis argues that building the symmetries of the theory directly into a neural network makes learning much more reliable than training a generic network on the same data. The first demonstration is translational symmetry: translation-equivariant convolutional networks predict the observables of a complex scalar field theory more accurately and with smaller models than strided or flattening architectures, and they generalize to lattice sizes and chemical potentials not seen in training. The main claim is the construction of lattice gauge equivariant convolutional neural networks (L-CNNs), whose layers are equivariant under local gauge transformations by design, and the demonstration that they can form Wilson loops of arbitrary size and regress traced loop observables and topological charge in 1+1D and 3+1D, where ordinary CNNs fail badly. A final chapter proposes neural gradient flow, a neural-network-driven ordinary differential equation on gauge links, as a way to generate lattice gauge configurations. If these results hold, lattice field theory practitioners can replace symmetry-agnostic networks with architectures that encode gauge invariance, reducing data and compute requirements for non-perturbative calculations.

What carries the argument

The central object is the lattice gauge equivariant convolutional network (L-CNN), and the identity that carries the argument is the lattice gauge equivariant convolution, $C_{x,i}(U,W) = \sum_{j,\mu,k} \omega_{i,j,\mu,k} U_{x,k\cdot\mu} W_{x+k\cdot\mu,j} U^\dagger_{x,k\cdot\mu}$, where $U_{x,k\cdot\mu}$ is the parallel transporter along lattice axis $\mu$. This parallel transport makes the layer equivariant under local gauge transformations and, with translation-invariant weights $\omega$, under translations as well. The bilinear layer multiplies locally transforming objects at the same site to combine loops, the trace layer projects to gauge-invariant outputs, and the exponentiation layer updates links while preserving unitarity and determinant; together these layers let a deep network build Wilson loops of arbitrary shape from elementary plaquettes.

What would settle it

Train a fixed-depth L-CNN to regress a contractible Wilson loop whose construction requires a connecting path with more axis-direction changes than the number of L-Conv layers available, such as a staircase loop in 3+1D; if even a wide network cannot match the accuracy it reaches on rectangular loops, the completeness step of the universality claim is false. A cheaper check is to enumerate all loops up to some perimeter and test whether each is expressible by the L-Conv/L-Bilin construction; one inexpressible contractible loop would refute the claim that arbitrary Wilson loops are generated.

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Extended reading notes

Core claim

The central result is that a network whose layers are gauge equivariant by construction is sufficient to express and learn gauge-invariant lattice observables. The paper defines L-CNNs built from lattice gauge equivariant convolutions (L-Conv), which parallel-transport local objects to a common site along straight lattice-axis paths before summing them with translation-invariant weights, combined with bilinear layers that multiply such objects and a trace layer that extracts gauge-invariant scalars. Repeated L-Conv and bilinear blocks can grow Wilson loops by doubling their area, and the paper gives a proof sketch, based on tessellating a loop into plaquettes, that any contractible Wilson loop can be generated this way; with Polyakov layers, non-contractible loops are covered too. Empirically, L-CNNs regress traced $1\times1$, $1\times2$, $2\times2$, and $4\times4$ Wilson loops in 1+1D and $2\times2$ and $4\times4$ loops plus the topological charge density in 3+1D with orders-of-magnitude lower mean squared error than gauge-symmetry-breaking CNNs with comparable parameter counts, and the trained models transfer to larger lattices. The scalar-field chapter establishes the analogous benefit for translational symmetry: an equivariant CNN outperforms strided and flattening baselines across tasks, sample sizes, and lattice sizes.

Load-bearing premise

The load-bearing premise is that every contractible Wilson loop can be decomposed by removing a unit plaquette whose connecting path from the loop origin is always reproducible by consecutive L-Conv layers; the proof sketch gives this as an induction step but does not prove that every required path and loop shape in D>2 is reachable with finite depth.

Editorial extensions

If this is right

  • L-CNNs can in principle approximate any gauge-equivariant function on the lattice, so Wilson-loop-based observables beyond those tested, such as improved actions, become learnable targets.
  • Because gauge invariance is exact by construction, training needs no gauge-copy data augmentation, and models trained on small lattices apply directly to larger lattices.
  • Translation-equivariant CNNs are not locked to the training lattice size, while flattening architectures are, so symmetry preservation removes a retraining bottleneck for lattice studies.
  • Neural gradient flow, if it trains successfully, turns configuration generation into a cheap integration of a learned differential equation from random initial links.
  • The failure mode of baseline CNNs grows with loop size—predictions collapse toward the training average—so symmetry is not a minor advantage but the deciding factor for these observables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The induction sketch in Section 3.4 is the fragile point: if some contractible loop requires a connecting path with more axis-direction changes than the available depth, the universality claim overreaches even though the tested rectangular-loop regressions stand as reported.
  • The same parallel-transport construction could be transplanted to non-Abelian gauge theories on curved lattices or manifolds by replacing straight-axis paths with geodesics, which would connect this method to equivariant networks on manifolds.
  • Neural gradient flow could be trained directly on expectation values rather than final link configurations, which would sidestep the need for paired initial-final samples and make the method usable where target configurations are not available.
  • The scalar-field results suggest that for theories with both global and local symmetries, the two equivariances compose cleanly: a translation-equivariant CNN after the trace layer inherits the full symmetry group of the lattice theory.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This thesis-style manuscript studies neural networks that respect the symmetries of lattice field theories. Chapter 2 formulates translation-equivariant CNNs for a complex scalar field in flux representation, comparing equivariant (EQ), strided (ST), and flattening (FL) architectures on three tasks: regression of the observables n and |phi|^2, detection of flux violations, and counting of open worms. Chapter 3 defines a family of gauge-equivariant layers (L-Conv, L-Bilin, L-Act, L-Exp, Trace, Plaq, Poly), assembles them into L-CNNs, sketches a proof that arbitrary Wilson loops can be generated by such networks, and reports regression experiments for traced Wilson loops and topological charge in 1+1D and 3+1D. Chapter 4 adapts neural ODEs to SU(N_c) gauge links, proposing a 'neural gradient flow' as a generative method and testing it on a single-SU(2)-link toy model. The central claim is that L-CNNs solve Wilson-loop regressions essentially perfectly while gauge-symmetry-breaking CNNs fail, demonstrating the advantage of building gauge equivariance into the architecture.

Significance. If the headline comparison were clean, the manuscript would provide a strong demonstration that hard-wiring gauge equivariance is a decisive inductive bias for lattice-gauge-theory observables, with practical implications for machine-learning-based lattice QCD. The layer-wise equivariance proofs in Sections 2.2 and 3.2 are clear and self-contained, and the experimental methodology is careful in several respects: optuna-based architecture search with validation-based selection, 10 to 50 random initializations per architecture, early stopping, and honest reporting of non-convergence and outliers. However, the central L-CNN-versus-CNN comparison in Chapter 3 is confounded by an asymmetry in the supervision target, and the Section 3.4 universality proof is a sketch rather than a rigorous argument. The neural-gradient-flow chapter is a suggestive proof of principle but does not yet demonstrate a practical sampling method. The empirical demonstration that L-CNNs can regress Wilson loops and topological charge to high accuracy is valuable on its own; the comparative claim against CNNs needs to be repaired before it can support the abstract's conclusion.

major comments (3)
  1. [§3.5.2 and §3.5.3] The headline claim that gauge-symmetric L-CNNs greatly outperform gauge-breaking CNNs rests on an asymmetric training and evaluation protocol. Section 3.5.2 states that the L-CNN output is defined at every lattice site and that omitting a global average pooling (GAP) layer 'leads to much easier convergence'; Section 3.5.3 states that CNNs 'without a GAP layer performed much worse' and that, to enable comparison, 'we take the lattice average over the final output layer of the L-CNN models.' Thus L-CNNs are trained with per-site MSE against the local labels W^{(m×n)}_{x,μν}, while the CNN with GAP and a dense head is trained against a global average label, and the L-CNN is only spatially averaged at evaluation time. A regressor trained on the global mean has no incentive to reproduce local spatial variation; the reported failure of CNNs on the negative values of W^{(1×2)} and W^{(2×2)} and the collapse to the training-set average for W^{(4×4)} are exactly the signatures of a mean predictor. This confound, rather than the presence or absence of gauge equivariance, can explain the observed gap. Please rerun the comparison with matched supervision: train the CNN with per-site outputs (for example, replace GAP by a 1×1 convolution at each site and use the same per-site MSE), or train the L-CNN with GAP and a global MSE, and report both configurations.
  2. [§3.4] The induction sketch does not establish the stated universality claim. The step 'By means of the iterated application of L-Convs, it is always possible to generate the path that connects the origin with the removed plaquette' is asserted but not proved. The L-Conv of Eq. (3.2.13) transports W only along straight, axis-aligned paths with translation-invariant weights; paths that turn corners require a specific composition of L-Conv and L-Bilin operations, and the sketch does not show that every path required by an arbitrary tessellation—nor every non-planar loop in D>2, where the surface tessellation is non-unique—is reachable with finite depth. Since the sentence 'L-CNNs can be seen as universal approximators of functions on the lattice' depends directly on this step, the section should either supply a rigorous constructive proof with explicit depth bounds or weaken the claim to the class of loops demonstrably constructible (for example, rectangular loops) and explicitly flag the unresolved general case.
  3. [§3.5.4] The hyperparameter schedules are not matched between the two architecture families in 1+1D. For CNNs the learning rate is 3×10^{-2}, the maximum number of epochs is 100, and the patience is 25; for L-CNNs the corresponding values are 3×10^{-3}, 20, and 5 for the 1×1 and 1×2 tasks, and 1×10^{-3}, 100, and 25 for the 2×2 and 4×4 tasks. Even if the supervision-target asymmetry of Section 3.5.3 is fixed, the reported performance gap could still be influenced by unequal training budgets. Please report results under a shared hyperparameter protocol, or justify why the per-architecture schedules are the appropriate fair comparison.
minor comments (6)
  1. [§3.2.9] In Eq. (3.2.52), the product defining the Polyakov loop should specify the range of k; with periodic boundary conditions the product over all k is ambiguous unless the number of links in the temporal direction is stated explicitly.
  2. [§3.5.5] The W^{(1×1)} comparison should be presented as a sanity check rather than as evidence of the gauge-equivariance advantage, because the L-CNN contains an explicit Trace layer and the input already contains the plaquette matrices; the large performance gap for this observable is partly by construction.
  3. [§3.5.7] Equation (3.5.6) defines QP, but Fig. 3.5 labels the ordinate Qplaq(τ); please align the notation.
  4. [§4.1] The derivation of the Hermiticity and tracelessness of H assumes time-independent gauge transformations; this assumption should be stated explicitly when deriving the transformation property in Eq. (4.1.28), since time-dependent gauge transformations would introduce additional terms.
  5. [§2.5] The observation that the observable n can be reproduced exactly by an equivariant linear model with a single 1×1 convolution should appear earlier in Chapter 2 and be used to frame Task I as a warm-up, so that the reader does not over-interpret the EQ advantage on that observable.
  6. [Abstract] The phrase 'L-CNNs are specifically designed ad hoc' is redundant; consider 'purpose-built' or simply 'designed'.

Circularity Check

3 steps flagged · score 6.0 of 10

Wilson-loop regressions reduce to the L-CNN's own loop-building construction; the CNN comparison remains independent.

  1. self definitional [Section 3.5.5, with Eq. (3.5.1), Eq. (3.2.46), and Section 3.2.1]
    "The first task we investigate is almost trivial: predicting the real value of the traced 1×1 Wilson loop as in Eq. (3.5.1). Since part of the input data consists of all possible 1×1 Wilson loops, the only required computations are first to perform the trace operation and then to take its real part."

    The target is W(1×1)_{x,μν} = (1/N_c) ReTr[U_{x,μν}] (Eq. 3.5.1), while the input tuple (U,W) already contains W = {U_{x,μν}} as the locally transforming variables (Section 3.2.1). The Trace layer (Eq. 3.2.46) outputs T_{x,i} = Tr[W_{x,i}]. Therefore the L-CNN 'prediction' of W(1×1) is the trace of an input plaquette, up to the fixed 1/N_c and Re operations. The reported L-CNN success on this regression is the architecture's definition, not an independent empirical finding; the paper's own 'almost trivial' comment confirms the reduction.

  2. self definitional [Section 3.3 and Section 3.5.2, with Eq. (3.2.13), Eq. (3.2.20), and the W(1×2), W(2×2), W(4×4) regressions]
    "Given that an L-CB layer doubles the maximum possible loop area, the repetition of n L-CB layers that take plaquettes as initial input allows to predict loops of area 2^n. ... it may seem trivial for L-CNNs to solve regression tasks involving Wilson loops, since they are intrinsically constructed to form arbitrarily sized loops."

    An m×n Wilson loop is a product of link variables along a rectangle; L-Conv (Eq. 3.2.13) parallel-transports plaquettes along the lattice axes and L-Bilin (Eq. 3.2.20) multiplies such locally transforming objects at a site. The sequence Plaq → L-Conv → L-Bilin is, by the authors' own description, exactly the mechanism that 'grows' a larger Wilson loop out of 1×1 plaquettes. Hence the 'success' on these regressions is substantially the execution of the loop-composition algorithm the layers were built to implement. The paper's own caveat that the task 'may seem trivial' because the networks are 'intrinsically constructed to form arbitrarily sized loops' states the reduction. The comparison with non-equivariant CNNs remains an independent benchmark, so the circularity is partial.

1 more flagged steps
  1. self definitional [Section 2.5, with Eq. (2.4.1) and Table 2.4]
    "This is not surprising if we consider that the observable n can be reproduced exactly by an equivariant linear model with a single 1×1 convolution."

    The target n is defined in Eq. (2.4.1) as n = (1/N_x N_t) Σ_x k_{x,2}, a linear function of one input channel. The EQ architecture in Table 2.4 begins with Conv(1×1,4,64) and ends with GlobalAvgPool + Linear; a single 1×1 convolution can select the k_{x,2} channel and the global average computes the sum. Thus the EQ model's success on the n part of Task I is guaranteed by the layer definitions. The paper explicitly labels this 'not surprising'; the |φ|^2 observable, which requires neighboring-site information, carries the non-trivial content of the comparison.

full rationale

The paper contains genuine circularity in the Wilson-loop benchmarks, but it is openly acknowledged and only partially load-bearing. The W(1×1) regression reduces to tracing an input plaquette: Eq. (3.5.1) defines the target as (1/N_c)ReTr of U_{x,μν}, and the input tuple already contains W = {U_{x,μν}}, with the Trace layer (Eq. 3.2.46) producing Tr[W]. The larger-loop regressions likewise use L-Conv (parallel transport) and L-Bilin (multiplication) in the exact combination the paper says can 'grow' arbitrarily sized Wilson loops; the paper's own caveat in Section 3.5.6 concedes the task 'may seem trivial... since they are intrinsically constructed to form arbitrarily sized loops.' The Chapter 2 scalar-field Task I contains a second, less central reduction: n is exactly reproducible by a single 1×1 equivariant convolution, as the paper states. None of these reductions, however, make the whole derivation circular. The headline comparison of L-CNNs against gauge-symmetry-breaking CNNs is an external benchmark with reported scatter plots and MSEs; the CNN failure on 4×4 loops is an independent empirical observation, even though the per-site vs GAP supervision asymmetry weakens it as a clean test of gauge symmetry. The universality claim is under-supported by the proof-by-induction sketch, but that is a completeness gap rather than a circular reduction. No load-bearing self-citation chain or imported uniqueness theorem was found; citations to the authors' own prior papers are used for background and for the 'number of loops grows exponentially' remark, not to force the central result. Overall score 6: some 'predictions' reduce by construction, but the central architecture comparison retains independent empirical content.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The thesis fits no physical constants: it is an ML-methods work. The only hand-chosen numerical scale is C = 100. The central claims rest on the dual/flux representation and the Wilson lattice action taken from the literature, on the non-Abelian Stokes theorem and CNN universality theorems, and on the paper-specific restriction of L-Conv to axis-aligned parallel transport. The introduced entities are the L-CNN layer family and the neural gradient flow proposal.

free parameters (1)
  • Topological charge rescaling constant C = 100
    Eq. (3.5.5) multiplies the topological charge density label by C = 100 to facilitate training; a hand-picked constant that changes the loss scale but not the physics.
assumptions (4)
  • domain assumption Duality (flux) representation of the complex scalar field with chemical potential, partition function Eq. (2.1.22), taken from Gattringer et al. [96]
    Chapter 2 defines all observables (n, |phi|^2, flux Fx) in this representation; the thesis cites [96] for the derivation rather than reproducing it.
  • standard math Non-Abelian Stokes' theorem: any contractible Wilson loop equals the ordered product over an enclosing surface, allowing tessellation into unit plaquettes
    Invoked in Section 3.4 as the foundation of the induction sketch that L-CNNs can generate arbitrary Wilson loops.
  • domain assumption Universal approximation theorems for CNNs transfer to the L-CNN stack (Trace layer plus ordinary CNN)
    Section 3.4 concludes L-CNN universality by combining the Wilson-loop generation claim with CNN universality [47,48]; the transfer to the gauge-equivariant setting is asserted, not proved.
  • ad hoc to paper Restricting parallel transport to straight axis-aligned paths (Eq. 3.2.13) does not limit the loop shapes constructible by L-CNNs
    Section 3.2.3 adopts axis-aligned straight-line transporters because unique lattice geodesics do not exist; the completeness of this restriction for arbitrary loop shapes is the load-bearing part of the Section 3.4 sketch.
invented entities (2)
  • L-CNN layer family (L-Conv, L-Bilin, L-Act, L-Exp, Trace, Plaq, Poly) independent evidence
    purpose: Gauge-equivariant neural network layers for lattice gauge theory, used for regression of Wilson loops and topological charge and for the neural gradient flow generator
    Layer equivariance is proven in Section 3.2 and the architecture is validated empirically in Section 3.5; subsequent independent works [84-86] adopt the construction, giving a falsifiable handle outside this thesis.
  • Neural gradient flow (NODE-based ODE on SU(Nc) links, Eq. 4.1.16)
    purpose: Proposed generative method to map random link configurations to configurations of a target theory, as the thesis says, 'propose it as a method to generate lattice gauge configurations'
    Tested only on a single-SU(2)-link toy model (Section 4.2); no full-lattice demonstration, acceptance metric, or autocorrelation benchmark is provided, so there is no falsifiable handle outside this paper yet.

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Cite this review

Pith. "Pith review of Symmetry-preserving neural networks in lattice field theories." pith.science (2026). https://pith.science/paper/3LQIFG25

@misc{pith2026250612493,
  author       = {Pith},
  title        = {Pith review of: Symmetry-preserving neural networks in lattice field theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LQIFG25}},
  note         = {Machine review of arXiv:2506.12493}
}
read the original abstract

This thesis deals with neural networks that respect symmetries and presents the advantages in applying them to lattice field theory problems. The concept of equivariance is explained, together with the reason why such a property is crucial for the network to preserve the desired symmetry. The benefits of choosing equivariant networks are first illustrated for translational symmetry on a complex scalar field toy model. The discussion is then extended to gauge theories, for which Lattice Gauge Equivariant Convolutional Neural Networks (L-CNNs) are specifically designed ad hoc. Regressions of physical observables such as Wilson loops are successfully solved by L-CNNs, whereas traditional architectures which are not gauge symmetric perform significantly worse. Finally, we introduce the technique of neural gradient flow, which is an ordinary differential equation solved by neural networks, and propose it as a method to generate lattice gauge configurations.

Figures

Figures reproduced from arXiv: 2506.12493 by the authors.

Figure 2.1
Figure 2.1. The three different architecture types chosen for the compari [PITH_FULL_IMAGE:figures/full_fig_p027_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. The pictures above illustrate six possible consecutive movements [PITH_FULL_IMAGE:figures/full_fig_p033_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Distributions of the field component kt . These histograms illus￾trate the distributions of kt in the training set (top), test set A (middle) and test set B (bottom). The test sets noticeably maintain a consistent distribu￾tion across the different lattice sizes. The training set and test set A feature mostly the same values of kt , although with largely different distributions. Test set B is characterized by a less… view at source ↗
Figures from the paper (27 more)
Figure 2.4
Figure 2.4. Figure 2.4: Distributions of f and ratio of W(f). The histograms depict the distributions of f in the training set (top histogram), test set A (middle histogram), and test set B (bottom histogram). Similar remarks can be made for the distribution of f in [PITH_FULL_IMAGE:figure…
Figure 2.5
Figure 2.5. Figure 2.5: Test loss on the test set of all 60 × 4 configurations as a function of the samples used to train. The top plot displays the results of the three architecture types trained without data augmentation. In the middle and bottom plot, the impact of training with data aug…
Figure 2.6
Figure 2.6. Figure 2.6: Predicted and true values for the ensemble averages [PITH_FULL_IMAGE:figures/full_fig_p044_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: True versus predicted observables for the best and the worst ST [PITH_FULL_IMAGE:figures/full_fig_p045_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: Overall test loss (top) and its contributions coming from each [PITH_FULL_IMAGE:figures/full_fig_p046_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Predicted and true ensemble averages of the physical observables [PITH_FULL_IMAGE:figures/full_fig_p048_2_9.png]
Figure 2.10
Figure 2.10. Figure 2.10: Scatter plot of predicted versus true observable values for the [PITH_FULL_IMAGE:figures/full_fig_p050_2_10.png]
Figure 2.11
Figure 2.11. Figure 2.11: Total test loss and its contributions given by the physical ob [PITH_FULL_IMAGE:figures/full_fig_p051_2_11.png]
Figure 2.12
Figure 2.12. Figure 2.12: Visualization of an open worm field configuration and of one of [PITH_FULL_IMAGE:figures/full_fig_p053_2_12.png]
Figure 2.13
Figure 2.13. Figure 2.13: Test loss (top) and test accuracy (bottom) of the best EQ [PITH_FULL_IMAGE:figures/full_fig_p057_2_13.png]
Figure 2.14
Figure 2.14. Figure 2.14: Test loss (top) and test accuracy (bottom) of the best EQ [PITH_FULL_IMAGE:figures/full_fig_p058_2_14.png]
Figure 2.15
Figure 2.15. Figure 2.15: Distributions of the link field kt . These two histograms feature the distributions of kt in the training set (top) and in the test set (bottom). Image from [1]. 2.6.2 Architecture search, training and testing An initial exploration phase is conducted to examine tre…
Figure 2.16
Figure 2.16. Figure 2.16: Distributions of the link field kx. These two histograms feature the distributions of kx in the training set (top) and in the test set (bottom). Image from [1]. and features a spatial pooling layer after each of them. An optional 1 × 1 convolution is inserted before…
Figure 2.17
Figure 2.17. Figure 2.17: Test loss (top) and test accuracy (bottom) of the architectures [PITH_FULL_IMAGE:figures/full_fig_p065_2_17.png]
Figure 2.18
Figure 2.18. Figure 2.18: Test loss (top) and test accuracy (bottom) of the architectures [PITH_FULL_IMAGE:figures/full_fig_p066_2_18.png]
Figure 2.19
Figure 2.19. Figure 2.19: Test loss on the 8 × 8 lattice versus validation loss for every instance of each architecture. The test loss is shown for all 20 instances of the winning architectures displayed in table 2.7. Training and validation were executed with Nworms ∈ {0, 5} and (η, µ) ∈ {(…
Figure 2.20
Figure 2.20. Figure 2.20: Test loss over the whole test set versus validation loss of every [PITH_FULL_IMAGE:figures/full_fig_p067_2_20.png]
Figure 3.1
Figure 3.1. Figure 3.1: An example of L-CNN architecture. Using gauge links [PITH_FULL_IMAGE:figures/full_fig_p085_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Sketch of the proof that L-CNNs can generate arbitrarily sized [PITH_FULL_IMAGE:figures/full_fig_p086_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Scatter plot showing predictions versus ground truth of best [PITH_FULL_IMAGE:figures/full_fig_p091_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Scatter plot showing predictions versus ground truth of best [PITH_FULL_IMAGE:figures/full_fig_p092_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Predictions of the best L-CNN model for the topological charge [PITH_FULL_IMAGE:figures/full_fig_p096_3_5.png]
Figure 4.1
Figure 4.1. Figure 4.1: Projection on the 3D unit sphere of 1 000 matrices taken from the [PITH_FULL_IMAGE:figures/full_fig_p105_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Test results. (a) Evolution of 30 samples projected on the 3D [PITH_FULL_IMAGE:figures/full_fig_p106_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Test results for final flow time τ = 10 based on training up to τ = 1. (a) Extrapolated evolution of 30 samples projected on the 3D unit sphere. The true and predicted trajectories lie too close to each other to be able to tell them apart. (b) Frobenius norm as a fun…
Figure 4.4
Figure 4.4. Figure 4.4: Mispredictions and their effect on the results. (a) We show that [PITH_FULL_IMAGE:figures/full_fig_p108_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Solution of an NODE of a U(1) element with adjoint sensitivity [PITH_FULL_IMAGE:figures/full_fig_p114_4_5.png]

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