Latent Dynamics · 2026-10-16
Same Target, Different Origins: Aligning Latent Rollout for Consistent PDE Emulation
Xinying Li, Jing Xu, Wenjun Wang, Limei Han, Fenglei Cao, Ruoxi Jiang, Yuan Qi
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TL;DR
ALR (Align Latent Rollout) is a latent-space regularization for autoregressive neural PDE solvers. It enforces cross-origin consistency: latent states and their local evolution responses should agree at a shared target time even when reached from different rollout origins/depths. Motivated by second-order Taylor analysis, ALR adds zeroth-order latent state consistency and first-order latent dynamics consistency via Jacobian–vector products, improving accuracy and reducing shared-target discrepancies.
Why it matters
Autoregressive neural PDE solvers can accumulate path-dependent errors, so predictions at the same target time can depend on rollout origin. ALR addresses this by explicitly coupling internal representations across origins at the shared target time—without changing the solver backbone—aiming to make learned evolution consistent with deterministic PDE flow.
Method
- Formulate cross-origin consistency for deterministic PDE evolution and define shared-target discrepancies in a chosen latent representation layer extracted from the neural operator.
- Propose ALR regularization: zeroth-order latent state consistency aligns latent states at the shared target time; first-order latent dynamics consistency aligns local induced evolution responses using Jacobian–vector products (stochastic JVPs).
- Integrate ALR into existing neural PDE backbones by regularizing suitable internal representations (no backbone architecture modifications).
Abstract (from arXiv)
Neural PDE solvers commonly generate multi-step predictions through autoregressive rollout, recursively applying a learned evolution operator and feeding each prediction back as input for the next step. Approximation errors can therefore propagate through the rollout, causing predictions at a shared target time to depend on their temporal origins and rollout depths. For a well-posed deterministic PDE flow, however, states at different times along the same physical trajectory should, under exact evolution, reach the same target state. To this end, we propose Align Latent Rollout (ALR), a regularization approach that promotes consistency across rollout origins through the neural PDE solver's internal representations. A second-order Taylor analysis of latent discrepancy propagation motivates two objectives: zeroth-order latent state consistency, which aligns representations at a shared target time, and first-order latent dynamics consistency, which aligns local evolution responses to shared perturbations. ALR can be integrated into different neural PDE solvers by regularizing suitable internal representations, without modifying their backbone architectures. Experiments across multiple PDE benchmarks and neural-operator backbones show that ALR improves prediction accuracy while reducing shared-target discrepancies across rollout origins.
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Publication record
Bibliographic evidence describes where and when a record appeared. It does not validate scientific claims.
First public release: 2026-10-15 · Indexed by Pub-AI: 2026-10-16 UTC
Same Target, Different Origins: Aligning Latent Rollout for Consistent PDE Emulation
Preprint posted: 2026-10-15 · Original record ↗
- arxiv
- 2610.21370
Explicit source versions (1)
- v1 · 2026-10-15
- Preprint available
Source date: 2026-10-15 · First detected: 2026-10-25 UTC · Historical baseline
Source evidence (1)
- Official evidence ↗ · atom:published (official metadata)
Abstract · arxiv
Neural PDE solvers commonly generate multi-step predictions through autoregressive rollout, recursively applying a learned evolution operator and feeding each prediction back as input for the next step. Approximation errors can therefore propagate through the rollout, causing predictions at a shared target time to depend on their temporal origins and rollout depths. For a well-posed deterministic PDE flow, however, states at different times along the same physical trajectory should, under exact evolution, reach the same target state. To this end, we propose Align Latent Rollout (ALR), a regularization approach that promotes consistency across rollout origins through the neural PDE solver's internal representations. A second-order Taylor analysis of latent discrepancy propagation motivates two objectives: zeroth-order latent state consistency, which aligns representations at a shared target time, and first-order latent dynamics consistency, which aligns local evolution responses to shared perturbations. ALR can be integrated into different neural PDE solvers by regularizing suitable internal representations, without modifying their backbone architectures. Experiments across multiple PDE benchmarks and neural-operator backbones show that ALR improves prediction accuracy while reducing shared-target discrepancies across rollout origins.