Latent Dynamics · 2026-10-07
Learning PDE solution operators with variable initial conditions via Latent Dynamics Networks
Stefano Maria Pizzamiglio, Stefano Pagani, Francesco Regazzoni
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TL;DR
The authors extend LDNet to handle PDEs with variable initial conditions while staying encoder-free and resolution-independent. They infer the initial latent state from early-time observations using either an auto-decoding latent-code optimization (AD-IC-LDNet) or a CAVIA-style meta-learning strategy (Meta-IC-LDNet). The coordinate-based decoder supports training from spatially subsampled data and recovering high-resolution fields at inference across multiple PDE phenomena.
Why it matters
Fixed-initial-condition latent-dynamics surrogates limit many-query and real-world scenarios where starting states vary. By treating latent-state initialization as an adaptation problem and keeping end-to-end encoder-free training, the approach aims to produce a resolution-independent surrogate operator for time-dependent PDEs from varying initial conditions, with faster adaptation via meta-learning.
Method
- Start from LDNet (neural ODE evolves a latent state; decoder maps latent state to PDE solution at any spatial position/time); LDNet is encoder-free and trained end-to-end.
- For variable initial conditions, define initial latent state s(0)=Ψ({y(ξ,τ)}_{ξ,τ in observed initial window}); use encoder-free strategies: (i) auto-decoder with per-example trainable initial latent codes optimized at inference, (ii) meta-learning where initial latent state is task-specific context and meta-parameters are shared across tasks.
- Use a coordinate-based decoder to allow training from spatially subsampled data and recovering high-resolution solution fields at inference; integrate neural ODE with a forward Euler scheme.
Limitation
The training of the Meta-IC-LDNet can be sensitive to the choice of hyperparameters and, in some applications, it requires a regularization strategy, chosen on a case-by-case basis, to reach a stable, well-conditioned latent space.
Abstract (from arXiv)
In many-query scenarios, data-driven surrogate models provide an efficient alternative to high-fidelity solvers for simulating physical systems governed by Partial Differential Equations (PDEs). In this context, the Latent Dynamics Network (LDNet) has recently demonstrated remarkable performance in predicting the response of spatio-temporal systems, combining Neural Ordinary Differential Equations with nonlinear dimensionality reduction. However, the original formulation assumes a fixed initial condition, limiting its applicability to many real-world applications where a system evolves from varying starting states. In this work, we overcome this limitation while keeping the end-to-end training procedure of the original LDNet and its encoder-free nature, which preserves its intrinsic independence from spatial resolution and grid topology. We infer the initial latent state directly from a small set of early-time observations, treating latent-state initialization as an adaptation problem, and investigate two strategies: an auto-decoding formulation and a meta-learning approach in which the initial latent state acts as a task-specific context variable. We demonstrate the accuracy of the proposed methods across diverse physical phenomena, spanning advection-diffusion, fluid dynamics, and solid mechanics. Meta-learning markedly accelerates latent-state inference and induces smoother, better-conditioned optimization landscapes, and spontaneously organizes the latent space into a structured representation that reflects physically meaningful features of the underlying dynamics. The coordinate-based decoder enables training from spatially subsampled data while recovering high-resolution solution fields at inference. The resulting approach provides an efficient and resolution-independent surrogate modeling framework for many-query simulations of time-dependent PDEs with varying initial conditions.
Related papers
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Publication record
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First public release: 2026-10-06 · Indexed by Pub-AI: 2026-10-07 UTC
Learning PDE solution operators with variable initial conditions via Latent Dynamics Networks
Preprint posted: 2026-10-06 · Original record ↗
- arxiv
- 2610.08475
Explicit source versions (1)
- v1 · 2026-10-06
- Preprint available
Source date: 2026-10-06 · First detected: 2026-10-25 UTC · Historical baseline
Source evidence (1)
- Official evidence ↗ · atom:published (official metadata)
Abstract · arxiv
In many-query scenarios, data-driven surrogate models provide an efficient alternative to high-fidelity solvers for simulating physical systems governed by Partial Differential Equations (PDEs). In this context, the Latent Dynamics Network (LDNet) has recently demonstrated remarkable performance in predicting the response of spatio-temporal systems, combining Neural Ordinary Differential Equations with nonlinear dimensionality reduction. However, the original formulation assumes a fixed initial condition, limiting its applicability to many real-world applications where a system evolves from varying starting states. In this work, we overcome this limitation while keeping the end-to-end training procedure of the original LDNet and its encoder-free nature, which preserves its intrinsic independence from spatial resolution and grid topology. We infer the initial latent state directly from a small set of early-time observations, treating latent-state initialization as an adaptation problem, and investigate two strategies: an auto-decoding formulation and a meta-learning approach in which the initial latent state acts as a task-specific context variable. We demonstrate the accuracy of the proposed methods across diverse physical phenomena, spanning advection-diffusion, fluid dynamics, and solid mechanics. Meta-learning markedly accelerates latent-state inference and induces smoother, better-conditioned optimization landscapes, and spontaneously organizes the latent space into a structured representation that reflects physically meaningful features of the underlying dynamics. The coordinate-based decoder enables training from spatially subsampled data while recovering high-resolution solution fields at inference. The resulting approach provides an efficient and resolution-independent surrogate modeling framework for many-query simulations of time-dependent PDEs with varying initial conditions.