Schr\"odinger-Inspired Time-Evolution for 4D Deformation Forecasting

2026-01-31

AI Summary

This work addresses the challenges of long-term instability and structural distortion in 4D (3D + time) spatiotemporal prediction by proposing a physics-informed neural architecture inspired by the Schrödinger equation. The method uniquely integrates a differentiable Schrödinger time-evolution operator end-to-end within a deep convolutional network, leveraging complex-valued wave functions to jointly learn amplitude (encoding structure), phase (encoding transport dynamics), and potential fields (modulating spatiotemporal interactions). This unified framework enables coherent prediction of both deformation fields and voxel intensities. Evaluated on synthetic benchmarks featuring complex deformations and topological changes, the approach significantly improves long-term prediction accuracy and stability while preserving anatomical consistency and mitigating error accumulation.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Deep Neural Architectures and Foundation Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 Schr\"odinger-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 $\psi = A e^{i\phi}$, which is evolved forward in time using a differentiable, unrolled Schr\"odinger 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 Schr\"odinger-type evolution operator, offering a principled pathway toward interpretable, stable, and anatomically consistent spatiotemporal prediction.

Problem

• 4D deformation forecasting
• spatiotemporal prediction
• physics-guided modeling
• temporal stability
• anatomical fidelity

Innovation

• Schrödinger-inspired
• physics-guided neural network
• 4D deformation forecasting
• time-evolution operator
• interpretable latent representation