cs.LGOct 24, 2026

Latent dynamics for reconstructing fine-scale PDE trajectories from coarse observations

Authors: Chenhao Si, Zishan Qian, Shing-Tung Yau, Rongling Wu, Ming Yan

Organizations: School of Data Science, The Chinese University of Hong Kong, Shenzhen, Shenzhen, China. · Institute of Statistics and Big Data, Renmin University, Beijing, China. · Beijing Key Laboratory of Topological Statistics and Applications for Complex Systems, Beijing Institute of Mathematical Sciences and Applications, Beijing, China. · Yau Mathematical Sciences Center, Tsinghua University, Beijing, China.

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Abstract

Coarse PDE observations omit spatial detail and intermediate states of evolving fields. We present Structured Latent Dynamics (SLD), a framework that reconstructs high-resolution trajectories from spatially and temporally downsampled simulations. SLD interpolates the input in time, adds a learned temporal correction and uses the resulting sequence to guide a hidden state on the coarse grid. Each update of the latent state combines a learned gated response with horizontal and vertical central differences and a five-point Laplacian of the state. The three spatial terms have learned channel-wise coefficients. Two Haar wavelet synthesis branches map the hidden state and refined coarse input to the high-resolution field. We evaluate SLD on two-dimensional Gray--Scott, complex Ginzburg--Landau and viscous Burgers systems, reconstructing 100 output frames from 25 input frames with eightfold refinement in each spatial direction. SLD achieves lower mean relative L2L_2 error and mean absolute error than unstructured latent dynamics and ConvLSTM baselines with comparable parameter counts, both on held-out trajectories and on later windows of trajectories used during training. SLD also reduces errors in regions with large reference gradients and across most or all of the evaluated later time intervals. A representative Gray--Scott comparison shows lower spectral error over most intermediate and high wavenumbers. Removing the fixed spatial response increased reconstruction error, supporting its contribution to the SLD architecture.

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Figure 1: Structured Latent Dynamics framework. a, A sequence of spatially coarse observations at sparse times is used to reconstruct a high-resolution field sequence. Information from the coarse sequence guides the reconstruction throughout the output interval. b, Temporal refinement. Linear interpolation increases the number of time positions. A temporal convolutional module predicts a correction at each spatial location, giving a refined sequence on the coarse grid. c, Structured latent dynamics. The hidden state is carried between output times. Its update combines a gated response computed from the state and the refined input with fixed horizontal and vertical central differences and a five-point Laplacian of the state. Learned channel-wise coefficients weight the three spatial responses. d, Multiscale wavelet synthesis. Two learned branches process the hidden state and the refined coarse field, respectively. Each predicts a complete set of Haar wavelet coefficients and reconstructs a high-resolution field by inverse wavelet synthesis. Their outputs are added to form the prediction.
Figure 2: Reconstruction from unseen initial conditions. a, Gray–Scott. b, Complex Ginzburg–Landau. c, Two-component viscous Burgers. For each system, high-resolution uu and vv reconstructions from SLD, UCLD and ConvLSTM are shown with the reference field. UCLD uses an unstructured convolutional latent vector field, and ConvLSTM uses a discrete recurrent update; both have parameter counts comparable to SLD. Yellow boxes mark the magnified regions. The right column shows validation relative L2L_{2} error histories under the shared training objective. Figure 3 reports the corresponding field errors, spectra and errors grouped by time and reference gradient magnitude.
Figure 3: Quantitative evaluation under unseen initial conditions. a, Relative L2L_{2} error and mean absolute error (MAE) for Gray–Scott (GS), complex Ginzburg–Landau (GL) and Burgers. Errors are first averaged over the test segments of each held-out trajectory. Bars and error bars show the mean and standard deviation across the ten held-out trajectories ( n=10n=10 ). b, Representative Gray–Scott comparison of the radial power spectrum E(k)E(k) and its relative error at wavenumber kk . These quantities measure the recovery of spatial variation at different scales. c, Mean relative L2L_{2} error for four test windows beginning at low-resolution indices 100, 150, 200 and 250. d, RMSE in ten groups of space–time points ordered by the local gradient magnitude of the reference field. The groups run from smooth to sharp regions, and shading marks the region with large gradients. SLD is compared with UCLD and ConvLSTM, which have comparable parameter counts.
Figure 4: Reconstruction in later temporal windows. a, Early segments S1S_{1} – S3S_{3} from all ten trajectories provide high-resolution targets for parameter optimization. Later segments S4S_{4} – S5S_{5} occupy disjoint time intervals. In each partition, later targets from five trajectories select the checkpoint and those from the other five evaluate the selected model. All models receive the low-resolution inputs of the later segments being reconstructed. b, Relative L2L_{2} error and mean absolute error for Gray–Scott (GS), complex Ginzburg–Landau (GL) and Burgers. Within each partition, errors are averaged over the two later segments from each of five test trajectories. Bars and error bars show the mean and s.d. of these summaries across five random partitions ( n=5n=5 ). c, Relative L2L_{2} error at each output time, with the S4S_{4} and S5S_{5} curves concatenated after the first ten high-resolution frames of each segment are omitted. Dashed vertical lines separate S4S_{4} and S5S_{5} . Shading shows the s.d. across the five curves, each obtained from one partition. UCLD and ConvLSTM have parameter counts comparable to SLD.
Figure 5: Additional Gray–Scott comparisons and ablation of the spatial response. a, Representative reconstructions of uu and vv from an unseen initial condition. Full SLD is compared with SLD trained with Sψ=0\mathcal{S}_{\psi}=0 and the additional baselines adapted to this task, as described in Supplementary Note 5.5 . b, Relative L2L_{2} error and mean absolute error under the same protocol for held-out trajectories. Bars and error bars show the mean and standard deviation across ten trajectories. The ablation retains the gated response, the remaining architecture and the training objective. Lower values indicate better reconstruction.
SymbolDescription
x1:Tin\mathbf{x}_{1:T_{\rm in}}Low-resolution input trajectory segment
y1:Tout\mathbf{y}_{1:T_{\rm out}}High-resolution target trajectory segment
y^1:Tout\widehat{\mathbf{y}}_{1:T_{\rm out}}Reconstructed high-resolution trajectory segment
TinT_{\rm in}Number of low-resolution input frames, Tin=25T_{\rm in}=25
ToutT_{\rm out}Number of high-resolution output frames, Tout=100T_{\rm out}=100
rrOutput-to-input sequence-length ratio, r=Tout/Tin=4r=T_{\rm out}/T_{\rm in}=4
Supplementary Table S1: Main notation used in the manuscript and Supplementary Information.
SystemSpatial DomainHR shapeLR shapeDownsampling
Gray–Scott[0,256)2[0,256)^{2}[2,3001,256,256][2,3001,256,256][2,751,32,32][2,751,32,32]8×8\times spatial, 4×4\times temporal
Ginzburg–Landau[−1,1)2[-1,1)^{2}[2,1200,256,256][2,1200,256,256][2,300,32,32][2,300,32,32]8×8\times spatial, 4×4\times temporal
Burgers[−1,1)2[-1,1)^{2}[2,1200,256,256][2,1200,256,256][2,300,32,32][2,300,32,32]8×8\times spatial, 4×4\times temporal
Supplementary Table S2: Summary of generated datasets. Each system contains ten trajectories from different initial-condition seeds. HR and LR denote high- and low-resolution trajectories.
SegmentLR start indexHR start indexUsage
S1S_{1}100400training/evaluation at early times
S2S_{2}150600training/evaluation at early times
S3S_{3}200800training/evaluation at early times
S4S_{4}225900evaluation in the first later window
S5S_{5}2501000evaluation in the second later window
Supplementary Table S3: Segment start indices used for reconstruction at later times. LR and HR denote the start indices in the low- and high-resolution trajectories. Each segment contains 25 LR frames and 100 HR frames. The protocol for unseen initial conditions uses starts 100, 150, 200 and 250.
ProtocolTraining dataValidation dataTest data
Unseen initial condition7 trajectories2 trajectories1 held-out trajectory
Later-time reconstructionS1S_{1} – S3S_{3} from all 10 trajS4S_{4} – S5S_{5} from val trajS4S_{4} – S5S_{5} from test traj
Supplementary Table S4: Summary of data partitioning in the two evaluation protocols. Partitions are defined by trajectory identity; the reported aggregation unit is specified separately for each protocol.
ComponentSetting
Input segment2525 LR frames, 2×32×322\times 32\times 32 per frame
Output segment100100 HR frames, 2×256×2562\times 256\times 256 per frame
Temporal upsampling4×4\times , linear interpolation plus temporal residual refinement
Spatial upsampling8×8\times , three-level Haar wavelet synthesis
Latent/guidance widthCh=64C_{h}=64
Guidance encodercircular 3×33\times 3 convolution, 2→642\to 64 channels
Supplementary Table S5: SLD architecture used in the main experiments. The same configuration is used for Gray–Scott, Ginzburg–Landau and Burgers systems and for all main evaluation protocols.
PDE systemMethodRel. L2L_{2} ↓\downarrowMAE ↓\downarrow
Gray–ScottSLD1.855×10−2±3.970×10−3\mathbf{1.855\times 10^{-2}\pm 3.970\times 10^{-3}}4.160×10−3±8.150×10−4\mathbf{4.160\times 10^{-3}\pm 8.150\times 10^{-4}}
 UCLD5.224×10−2±1.005×10−25.224\times 10^{-2}\pm 1.005\times 10^{-2}1.570×10−2±3.284×10−31.570\times 10^{-2}\pm 3.284\times 10^{-3}
 ConvLSTM4.550×10−2±5.852×10−34.550\times 10^{-2}\pm 5.852\times 10^{-3}1.048×10−2±1.427×10−31.048\times 10^{-2}\pm 1.427\times 10^{-3}
Ginzburg–LandauSLD1.820×10−2±5.127×10−3\mathbf{1.820\times 10^{-2}\pm 5.127\times 10^{-3}}6.017×10−3±1.866×10−3\mathbf{6.017\times 10^{-3}\pm 1.866\times 10^{-3}}
 UCLD4.347×10−2±3.348×10−34.347\times 10^{-2}\pm 3.348\times 10^{-3}1.390×10−2±7.690×10−41.390\times 10^{-2}\pm 7.690\times 10^{-4}
 ConvLSTM4.561×10−2±2.439×10−34.561\times 10^{-2}\pm 2.439\times 10^{-3}1.189×10−2±5.516×10−41.189\times 10^{-2}\pm 5.516\times 10^{-4}
Supplementary Table S6: Numerical results for reconstruction from unseen initial conditions. Rel. L2L_{2} denotes relative L2L_{2} error and MAE denotes mean absolute error; lower values indicate better reconstruction. Errors are averaged over the four segments of each held-out trajectory. Values are mean ±\pm s.d. over the ten resulting trajectory means ( n=10n=10 ), with every trajectory serving once as the test identity.
PDE systemMethodRel. L2L_{2} ↓\downarrowMAE ↓\downarrow
Gray–ScottSLD1.574×10−2±8.054×10−4\mathbf{1.574\times 10^{-2}\pm 8.054\times 10^{-4}}3.593×10−3±2.011×10−4\mathbf{3.593\times 10^{-3}\pm 2.011\times 10^{-4}}
 UCLD8.556×10−2±1.386×10−28.556\times 10^{-2}\pm 1.386\times 10^{-2}2.997×10−2±5.108×10−32.997\times 10^{-2}\pm 5.108\times 10^{-3}
 ConvLSTM3.613×10−2±8.396×10−43.613\times 10^{-2}\pm 8.396\times 10^{-4}7.554×10−3±2.409×10−47.554\times 10^{-3}\pm 2.409\times 10^{-4}
Ginzburg–LandauSLD1.250×10−2±4.904×10−4\mathbf{1.250\times 10^{-2}\pm 4.904\times 10^{-4}}3.765×10−3±1.002×10−4\mathbf{3.765\times 10^{-3}\pm 1.002\times 10^{-4}}
 UCLD3.703×10−2±4.419×10−33.703\times 10^{-2}\pm 4.419\times 10^{-3}1.345×10−2±1.844×10−31.345\times 10^{-2}\pm 1.844\times 10^{-3}
 ConvLSTM4.575×10−2±3.246×10−44.575\times 10^{-2}\pm 3.246\times 10^{-4}1.122×10−2±1.171×10−41.122\times 10^{-2}\pm 1.171\times 10^{-4}
Supplementary Table S7: Numerical results for reconstruction at later times. Rel. L2L_{2} denotes relative L2L_{2} error and MAE denotes mean absolute error; lower values indicate better reconstruction. Within each partition, errors are averaged over the two later segments from each of five test trajectories. Values are mean ±\pm s.d. of the five partition means ( n=5n=5 repeated partitions).
MethodRel. L2L_{2} ↓\downarrowMAE ↓\downarrow
SLD1.855×10−2±3.970×10−3\mathbf{1.855\times 10^{-2}\pm 3.970\times 10^{-3}}4.160×10−3±8.150×10−4\mathbf{4.160\times 10^{-3}\pm 8.150\times 10^{-4}}
SLD without local response3.720×10−2±1.433×10−33.720\times 10^{-2}\pm 1.433\times 10^{-3}7.825×10−3±2.621×10−47.825\times 10^{-3}\pm 2.621\times 10^{-4}
FNO1.011×10−1±6.480×10−31.011\times 10^{-1}\pm 6.480\times 10^{-3}3.090×10−2±2.057×10−33.090\times 10^{-2}\pm 2.057\times 10^{-3}
LNO1.056×10−1±2.224×10−21.056\times 10^{-1}\pm 2.224\times 10^{-2}3.322×10−2±7.878×10−33.322\times 10^{-2}\pm 7.878\times 10^{-3}
MambaNO5.607×10−2±8.965×10−35.607\times 10^{-2}\pm 8.965\times 10^{-3}1.636×10−2±3.496×10−31.636\times 10^{-2}\pm 3.496\times 10^{-3}
OSEDiff6.487×10−2±3.849×10−36.487\times 10^{-2}\pm 3.849\times 10^{-3}1.970×10−2±1.237×10−31.970\times 10^{-2}\pm 1.237\times 10^{-3}
Supplementary Table S8: Numerical results for the additional Gray–Scott baselines and the local response ablation under the protocol for unseen initial conditions. Rel. L2L_{2} denotes relative L2L_{2} error and MAE denotes mean absolute error; lower values indicate better reconstruction. Values are mean ±\pm s.d. across ten held-out trajectories after averaging the four test segments within each trajectory.
Supplementary Figure S1: Additional Gray–Scott reconstructions for unseen initial conditions. SLD reconstructions of the two state variables and the corresponding error maps for selected held-out examples.
Supplementary Figure S2: Additional complex Ginzburg–Landau reconstructions for unseen initial conditions. SLD reconstructions of the two state variables and the corresponding error maps for selected held-out examples.
Supplementary Figure S3: Additional Burgers reconstructions for unseen initial conditions. SLD reconstructions of the two state variables and the corresponding error maps for selected held-out examples.
Supplementary Figure S4: Spatial maps of absolute reconstruction error. a, Gray–Scott. b, Complex Ginzburg–Landau. c, Burgers. Columns compare SLD, UCLD and ConvLSTM, and rows correspond to the two state variables.
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