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hide: - toc description: The registered physical systems reconstructed by the same recipe: overview figure, per-system comparisons and optimization animations from one gallery run on a CPU.

Gallery

The registered inverse problems, 1-D, 2-D and volumetric, linear and nonlinear, elliptic, hyperbolic, parabolic and spectral, are all solved by the same recipe: a coordinate neural field, the instrument's physics as a hard constraint, a coarse-to-fine curriculum. Every measurement is simulated with an independent, finer model than the one being inverted (no inverse crime), so each reconstruction has to cope with model error as well as noise.

Gallery overview of the twelve 1-D and 2-D systems, each shown as ground truth, measurement and reconstruction with its metric, solve time and step count

The twelve 1-D and 2-D systems as ground truth · measurement · reconstruction, from one run of examples/gallery.py (smoke presets at 0.2 × the default budgets, CPU, 182 s for all systems). The five volumetric systems are on the 3-D showcase; the full overview of all systems opens at full resolution. Click any figure to zoom.

Legend

Numbers from the run's manifest (runs/gallery/physics/manifest.json): smoke preset, one seed, 0.2 × the default step budget (× 2 for the volumetric systems), CPU with 4 threads, torch 2.11. They show what a few seconds of CPU buy; the instance pages report converged and paper-scale numbers.

system recovered from grid · steps · solve result
toy1d a 1-D signal from 64 blurred samples 64 · 180 · 0.6 s PSNR 25.1 dB
NV relaxometry · NeTMY spin sources from noise spectra at 50 frequencies 24² · 600 · 0.6 s Hungarian F1 1.00
Thermal tomography · NeFTY 3-D diffusivity from 30 surface frames 16²×6 · 600 · 6.1 s IoU 0.98 · PSNR 10.8 dB
Deconvolution an image from its blurred copy 32² · 450 · 0.4 s PSNR 28.1 dB
Sparse-view CT attenuation from 16 projections 32² · 450 · 0.4 s PSNR 25.5 dB
Poisson source sources from 30 % of the potential 32² · 600 · 0.6 s PSNR 30.7 dB
EIT conductivity from boundary voltages, 12 patterns 24² · 600 · 5.3 s PSNR 23.5 dB · IoU 0.94
Darcy flow log-permeability from well gauges, 4 configurations 16² · 400 · 3.9 s PSNR 24.3 dB
Current density a sheet current from its stray field 32² · 600 · 0.7 s current PSNR 29.7 dB
Full-waveform inversion sound speed from 6 × 32 acoustic traces 24² · 400 · 14.4 s PSNR 26.6 dB
Diffraction tomography contrast from complex fields, 16 angles 32² · 500 · 0.7 s PSNR 35.7 dB
Holography phase from intensities at 3 distances 32² · 400 · 0.6 s PSNR 35.0 dB
Reaction–diffusion the feed rate from 4 snapshots of 2 species 32² · 250 · 4.3 s PSNR 30.2 dB
3-D CT attenuation from 12 views of every slice 32²×16 · 800 · 7.3 s PSNR 29.4 dB · refined IoU 0.75 → 0.80
3-D deconvolution fluorophores from a blurred z-stack 32²×16 · 800 · 12.5 s PSNR 26.5 dB · refined IoU 0.65 → 0.71
Diffuse optical tomography absorption from surface reflectance, 9 sources 20²×10 · 700 · 17.9 s PSNR 28.5 dB · refined IoU 0.71 → 0.74
Photoacoustic tomography initial pressure from 64 sensor traces 20²×14 · 400 · 8.4 s PSNR 19.6 dB · refined IoU 0.73 → 0.80

The PSNR of thermal tomography is low because the bulk diffusivity is biased: with the smoke preset's coarse implicit time step the inversion compensates model error by raising α (the model-error floor, see Limitations and caveats). The defect itself is recovered (IoU 0.98).

System by system

Each figure shows the measurement, the ground truth, the reconstruction (with its metrics) and the signed error; the volumetric systems show depth slices, cross-sections through the anomaly and a projection. The same figures open every instance page.

NV relaxometry: summed noise spectrum, ground-truth spin sources, recovered sources with matched peaks, signed error

NV relaxometry: the Larmor field on the recovered support

Thermal tomography: mean of 30 surface frames, ground-truth depth slices, smooth reconstruction and the refused refinement, with cross-sections and projections

toy1d: ground truth, measurement and reconstruction of a 1-D signal

Deconvolution: blurred image, ground truth, reconstruction, signed error

Sparse-view CT: sinogram, Shepp-Logan ground truth, reconstruction, signed error

Poisson source: sparse potential observations, sources, reconstruction, signed error

EIT: boundary data, conductivity, reconstruction, signed error

Darcy flow: pressure sensors, log-permeability, reconstruction, signed error

Current density: stray field image, current magnitude, reconstruction, signed error

Full-waveform inversion: a shot gather, sound speed, reconstruction, signed error

Diffraction tomography: complex scattered field, contrast, reconstruction, signed error

Holography: intensity at the first distance, phase, reconstruction, signed error

Reaction-diffusion: mean snapshot, feed-rate field, reconstruction, signed error

3-D CT: sinogram, depth slices of ground truth and reconstruction, cross-sections, projection

3-D deconvolution: blurred stack, depth slices, cross-sections, projection

Diffuse optical tomography: surface data, depth slices, cross-sections, projection

Photoacoustic tomography: sensor traces, depth slices, cross-sections, projection

The 3-D showcase adds isosurfaces and interactive viewers for these systems.

Optimization in motion

The field during optimization — the coarse stage finds the structure, the fine stage adds detail after the annealed frequency bands reopen (the red dot marks the step on the loss curve).

3-D CT: four depth slices of the reconstruction converging toward the ground truth Full-waveform inversion: the sound-speed map converging over two stages Current density: the current magnitude converging NV relaxometry: sparse sources emerging from a uniform start

python examples/gallery.py --budget 0.2 --out runs/gallery --html    # CPU, about 3 minutes
python examples/gallery.py --budget 1.0 --device cuda --out runs/gallery --html   # full budgets

--html writes a self-contained index.html report with every figure and the interactive viewers. See Visualization for the figure types and the options.