Article

Reconstruction

What one frozen backbone reconstructs

The analysis backbone of Daedalus is trained exclusively on radial sines on a disk. This article examines how it reconstructs inputs outside that distribution, including shocks, waves, turbulent flows, stresses on bodies with holes and photographs, how its error decreases with the partition count, how many points a partition requires, and what it returns from a curve or from part of a field.

Based on Section 3 and Appendix B of the paper

Physical fields from nine datasets

The paper tests the analysis backbone on ten fields from real datasets, seven fields on regular grids, of compressible flow with shocks and shear layers, of an acoustic wave and of an incompressible flow, encoded with 1024 partitions, and three fields of elastic bodies on point clouds with holes, encoded with 128 partitions. None of these fields is a radial sine or is defined on a disk, and the domains of the elastic bodies are not convex. The reconstructions recover fine texture in addition to the coarse layout.

Ten pairs of fields and reconstructions: shocks, an acoustic wave, turbulent velocity, shear layers, and stress and displacement fields on bodies with holes, each pair labeled with its error
Reconstruction across domains, from a synthetic-only model. Every field is followed directly to its right by its reconstruction, on the same color scale, with the relative L2 error of the reconstruction in its top-left corner. Top row, the compressible-Euler density of CEU-RiemannKH, the acoustic wave of AWA-Layer, the incompressible-flow velocity of IEU-PwC, and the density of CEU-Gauss and of CEU-Riemann. Bottom row, the density of CEU-RiemannCurved and of CEU-KH, the displacement uy of one sample and the Cauchy stress σyy of another of SEQ-CloudCircle-Hyp, on a domain with a hole, and the shear stress σxy of SEQ-SquareCloud-Lin, a linear-elastic body on a square domain with two holes.

The geometry of a domain reaches the backbone only through the partitioner. A grid of pixels, a disk and the point cloud of a body with holes are all cut into small partitions, each of which is close to convex in its own frame, so an unseen domain poses no difficulty of its own. The accuracy is governed by the variation of the field within each partition, which is independent of the global domain.

Photographs

A photograph is three scalar functions on a pixel grid, and the frozen model encodes each color channel like any other input. The paper reconstructs 48 photographs in four groups, animals, minerals and crystals, paintings, and Persian art and places, most of them Featured Pictures of Wikimedia Commons. Each is cut to its central square, resampled to 512 × 512 pixels and reconstructed with 4096 partitions per channel, about 64 pixels per partition, and the median peak signal-to-noise ratio (PSNR) is 42.3 dB, with a range from 29.6 to 52.3 dB. The lowest values belong to texture finer than a partition, such as a densely painted fresco, a pointillist painting or rhodochrosite crystals on a granular rock, and the highest to content dominated by smooth regions, such as whales in open water or a clownfish before a blurred anemone.

Twelve photographs of animals, each beside its reconstruction with its PSNR Twelve photographs of minerals and crystals, each beside its reconstruction with its PSNR Twelve paintings, each beside its reconstruction with its PSNR Twelve photographs of Persian art and places, each beside its reconstruction with its PSNR
The four groups of the paper, from top to bottom animals, minerals and crystals, paintings, and Persian art and places. Every photograph, cut to its central square, is followed directly to its right by its reconstruction, with its PSNR in the top-left corner. Photographs from Wikimedia Commons.

The partition count as a resolution

The partition count is the resolution control of the representation. In every case examined in the paper, the error decreases steadily as the count grows. For the Cauchy stress σyy of a hyperelastic body on a domain with a hole it falls from 16.0% with a single partition to 0.82% with 256. The error concentrates around the hole and at the corners of the boundary, where the stress varies fastest, and recedes there as the partitions shrink. An incompressible-flow velocity follows the same course, from 36.3% with one partition to 0.18% with 256.

A stress field on a body with a hole reconstructed with 1, 4, 16, 64 and 256 partitions, with the error maps below fading
The Cauchy normal stress σyy of a hyperelastic body on a domain with a hole, reconstructed with 1 to 256 partitions (top row, with the relative L2 error above every reconstruction and the ground truth on the right) and the absolute error (bottom row).

Points required per partition

The partition count fixes how many embeddings describe a field, and the input resolution how many points each embedding is computed from. Growing the partition count on a fixed cloud eventually starves every partition of points. Scored at points the encoder did not see, on a cloud of 8192 points, the error is lowest at 64 partitions of 128 points and grows almost fivefold by 512 partitions of 16 points. At the encoded points themselves the error would continue to fall, since an embedding can store the few values it receives, and the loss of accuracy is visible only at held-out points. Growing the cloud with the partition count keeps the points per partition constant, and the held-out error then keeps falling at every density tried, from 16 to 128 points per partition, to between 1.34% and 0.10% at 2048 partitions, the denser partitions lower throughout. More partitions therefore cost accuracy only when they leave too few points in each.

Points per partition. The median relative L2 error at points the encoder did not see, on the evaluation sines, against the partition count. With the cloud fixed at 8192 points (orange), more partitions leave fewer points in each, and past 64 partitions the error grows. With the cloud grown in proportion to the partitions, at 16 to 128 points per partition (green, the denser, the stronger the line), the error keeps falling. The chips under the chart show or hide each line.

The widget below shows one function of that experiment, decoded at the pixels of the disk, none of which the encoder saw.

Encoder's points and partition centers

Decoded at held-out pixels

Error absolute

error at held-out pixelserror at the encoder's pointscurrent countrelative L², log scale, against the partitions
Partitions
64
points per partition – error at held-out pixels – error at the encoder's points –
Held-out points, pre-computed with the analysis backbone. One function of the evaluation sines is encoded from the same 8192 points (small dots, a quarter of them shown) with 1 to 512 partitions (yellow centers), without overlap, and decoded at the pixels of the disk, none of which the encoder saw. The curve gives the error at those pixels (green) and at the encoder's own points (gray, dashed). Up to about 16 partitions both decrease together. The held-out error then levels off and, beyond 64, where a partition holds too few points to determine the field between them, increases again, while the error at the encoder's points continues to fall.

Functions on curves

The encoder aggregates whatever points a partition holds, so a partition that contains only a piece of a curve is as admissible an input as one that fills an area. Although trained only on functions that fill a disk, the same model accepts a function sampled on a one-dimensional curve in the plane, such as a boundary trace. Encoded with 16 partitions and decoded on the same points, the reconstruction follows the observed values along each curve with a small, unstructured error.

Four curves carrying a function, their reconstruction and the absolute error along them
Functions on curves. Four smooth random curves in the plane (yellow), each carrying a scalar function drawn as a band colored by its value (left of each triple), reconstructed with 16 partitions (middle), and the absolute error, averaged over 17 neighboring points along the curve (right).

A curve under-determines the field around it, yet the decoder, which can be queried anywhere, returns a two-dimensional function. Observed on a straight line through the disk, a field is tracked along the line, while off the line the decoder returns a plausible smooth continuation that the observation does not constrain, a different one for every direction of the line. Different curves through one field likewise give different plausible completions, which indicates that the encoder does not return a memorized training function, and as more curves cover the domain the observations constrain more of it and the true field re-emerges. A latent space shared by functions on a domain and on its boundary serves operators whose inputs and outputs live on sets of different dimension, since boundary data then arrive already extended into the interior.

One field observed on straight lines at four angles, the truth above and the field inferred from each line below Six different curves through one field, each giving a different completed field
One-dimensional inputs. Top, one field (truth) observed on straight lines through the disk at four angles (yellow) and the field decoded from each line on a single partition (inferred): along the line the decoded field tracks the truth, and off the line the decoder fills in a plausible field. Bottom, six curves through one underlying field give six different plausible completions.
One field and its reconstruction from an increasing number of curves, the field re-emerging as the curves cover the disk
Coverage by curves. One underlying field (left) observed on an increasing number of curves (yellow). The more of the domain the curves cover, the more of the field the observations constrain.

Partial observations

Two experiments on a single partition probe partial observation from opposite sides, for radial-sine fields with 1 to 10 modes in the training band and a mask at radius 0.5. Observing only the inner disk and decoding on the whole disk, the observed region is reconstructed to below 1%, the continuation stays at about 5% in the band 0.5 < r < 0.6, and it becomes unconstrained further out, reaching 44 to 77% in the outermost band. Observing only the annulus 0.5 < r < 1 instead, the inner disk is filled in, an inpainting task, to about 2% for one or two radial modes, and the error grows to about 20% for five and ten modes, as the unobserved region becomes increasingly under-determined.

Truth observed part lit

Decoded from one partition

Error absolute

Observed
Field
Mask radius
0.5
error on the observed part – error on the unobserved part –
error on the unobserved parterror on the observed partrelative L², log scale, against the mask radius
Partial observations, pre-computed with the analysis backbone. A field of two radial-sine modes is observed at 1024 points inside the dashed circle (extrapolation) or outside it (inpainting), encoded as one partition, and decoded on the whole disk. At f ∈ [0.2, 1.5] the observed part is reconstructed to below 3% at every mask, inpainting fills a small hole closely, and extrapolation degrades with the distance from the observation. At f ∈ [1.5, 3] one partition holds many oscillations, its error on the observed part grows with the observed disk in extrapolation and as the observed ring thins in inpainting, and the completion is a plausible field of the training family rather than the truth. The chart gives both errors at every mask radius of the chosen setting (solid, unobserved; dashed, observed), the current radius ringed.

Where an observation leaves part of a field unconstrained, the completion resembles the training functions, a smooth field off a curve and radial wavefronts beyond an observed disk. This bias is a property of the training distribution rather than of the architecture. A backbone trained on another family of functions would complete an unobserved region with functions of that family, so the training distribution acts as the prior of every partially observed reconstruction and should match the structure an application expects where it observes nothing.

Two live demos apply the same single-partition completion to inputs of one's own choosing. The inpainting demo hides holes in a field or a picture, painted by hand or taken from a set of preset masks, and fills them from the embedding of the observed rest. The outpainting demo observes only a window, of adjustable shape and position, and continues the field from it over the whole disk.

Inpainting demo → Outpainting demo →