Schrödinger-Inspired Time-Evolution for 4D Deformation Forecasting

TL;DR

The paper introduces a Schrödinger-inspired, physics-guided neural framework for 4D deformation forecasting that encodes observed volumes into a complex latent state $\psi=A\,e^{i\Phi}$ driven by a potential $V$ and evolved via an approximate Crank–Nicolson scheme. By unrolling $N$ time steps, the model forecasts $\hat{X}_{t+1}=|\psi^{(N)}|^2$ and optimizes a loss combining $\ell_2$ reconstruction with total-variation regularization, achieving improved long-horizon stability and anatomically faithful predictions. The latent fields provide interpretability, with $A$ capturing morphology, $\Phi$ encoding transport, and $V$ acting as a deformation driver, as demonstrated on synthetic 4D/3D datasets and extended to 2D. Across ablations, a moderate unroll depth (around $N=50$) balances accuracy, detail preservation, and computational efficiency, while excessive unrolling causes drift and degraded surface fidelity. The approach offers a principled pathway toward stable, interpretable 4D spatiotemporal forecasting with potential applications in MR-guided radiotherapy, material dynamics, and geophysical deformation modeling.

Abstract

Spatiotemporal forecasting of complex three-dimensional phenomena (4D: 3D + time) is fundamental to applications in medical imaging, fluid and material dynamics, and geophysics. In contrast to unconstrained neural forecasting models, we propose a Schrödinger-inspired, physics-guided neural architecture that embeds an explicit time-evolution operator within a deep convolutional framework for 4D prediction. From observed volumetric sequences, the model learns voxelwise amplitude, phase, and potential fields that define a complex-valued wavefunction $ψ= A e^{iφ}$, which is evolved forward in time using a differentiable, unrolled Schrödinger time stepper. This physics-guided formulation yields several key advantages: (i) temporal stability arising from the structured evolution operator, which mitigates drift and error accumulation in long-horizon forecasting; (ii) an interpretable latent representation, where phase encodes transport dynamics, amplitude captures structural intensity, and the learned potential governs spatiotemporal interactions; and (iii) natural compatibility with deformation-based synthesis, which is critical for preserving anatomical fidelity in medical imaging applications. By integrating physical priors directly into the learning process, the proposed approach combines the expressivity of deep networks with the robustness and interpretability of physics-based modeling. We demonstrate accurate and stable prediction of future 4D states, including volumetric intensities and deformation fields, on synthetic benchmarks that emulate realistic shape deformations and topological changes. To our knowledge, this is the first end-to-end 4D neural forecasting framework to incorporate a Schrödinger-type evolution operator, offering a principled pathway toward interpretable, stable, and anatomically consistent spatiotemporal prediction.

Figures (10)

• Figure 1: Examples of 2D dataset samples at different timepoints, illustrating volumetric irregularity and realistic deformation dynamics.
• Figure 2: Isosurface visualization of random 3D dataset sample at different timepoint, showing diverse tumor-like morphologies with irregular boundaries.
• Figure 3: Dice--threshold sweep for different unroll steps. Curves illustrate operating characteristics and robustness to the choice of binarization threshold $\tau$.
• Figure 4: Visual comparison of forecasted and ground truth deformations at $t=5$ using 50 unroll steps with $\Delta t=0.02$. Top row: last observed input ($t=4$). Second row: forecasted deformation. Third row: ground truth at $t=5$.
• Figure 5: Distribution of voxelwise forecasting errors across the test set. The error distribution is centered near zero with most errors confined to $\pm 0.004$, confirming unbiased and stable predictions.