Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations

• URL: http://arxiv.org/abs/2605.13790v1
• Date: Wed, 13 May 2026 17:11:07 GMT
• Title: Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations
• Abstract summary: Di-BiLPS is a unified neural framework that handles both forward and inverse PDE problems under extremely sparse observations.<n>It consistently achieves SOTA performance under extremely sparse inputs, while substantially reducing computational cost.<n>It enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.
• Score: 3.523696436541849
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

Related papers

• Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems [48.15157016272698]
  latent-space diffusion posterior sampling (L-DPS) is an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs)<n>L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines in sparse and noisy regimes.
  arXiv  Detail & Related papers  (2026-06-25T04:33:44Z)
• Kinetic-based regularization: Learning spatial derivatives and PDE applications [0.0]
  We extend kinetic-based regularization to learn spatial derivatives with provable second-order accuracy in 1D.<n>The fully localized formulation enables efficient, noise-adaptive derivative estimation without requiring global system solving or smoothing.<n>Preliminary results show that coupling KBR with conservative solvers enables stable shock capture in 1D hyperbolic PDEs.
  arXiv  Detail & Related papers  (2026-03-06T15:32:27Z)
• LEVDA: Latent Ensemble Variational Data Assimilation via Differentiable Dynamics [6.953554594702111]
  We propose Latent Ensemble Data Assimilation (LEVDA), an ensemble-space variational smoother.<n>It assimilates states and unknown parameters without the need for adjoint code or auxiliary observation-to-latent encoders.<n>It substantially improved assimilation accuracy and computational efficiency compared to full-state 4DEnVar.
  arXiv  Detail & Related papers  (2026-02-23T00:54:59Z)
• Parallel Diffusion Solver via Residual Dirichlet Policy Optimization [88.7827307535107]
  Diffusion models (DMs) have achieved state-of-the-art generative performance but suffer from high sampling latency due to their sequential denoising nature.<n>Existing solver-based acceleration methods often face significant image quality degradation under a low-dimensional budget.<n>We propose the Ensemble Parallel Direction solver (dubbed as EPD-EPr), a novel ODE solver that mitigates these errors by incorporating multiple gradient parallel evaluations in each step.
  arXiv  Detail & Related papers  (2025-12-28T05:48:55Z)
• BiLO: Bilevel Local Operator Learning for PDE Inverse Problems. Part II: Efficient Uncertainty Quantification with Low-Rank Adaptation [9.229577043169224]
  Uncertainty quantification and inverse problems governed by partial differential equations (PDEs) are central to a wide range of scientific and engineering applications.<n>We extend Bilevel Local Operator Learning (BiLO) for PDE-constrained optimization problems developed in Part 1 to the Bayesian inference framework.
  arXiv  Detail & Related papers  (2025-07-22T21:20:20Z)
• Latent Mamba Operator for Partial Differential Equations [8.410938527671341]
  We introduce the Latent Mamba Operator (LaMO), which integrates the efficiency of state-space models (SSMs) in latent space with the expressive power of kernel integral formulations in neural operators.<n>LaMOs achieve consistent state-of-the-art (SOTA) performance, with a 32.3% improvement over existing baselines in solution operator approximation.
  arXiv  Detail & Related papers  (2025-05-25T11:51:31Z)
• Guided Diffusion Sampling on Function Spaces with Applications to PDEs [112.09025802445329]
  We propose a general framework for conditional sampling in PDE-based inverse problems.<n>This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning.<n>Our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines.
  arXiv  Detail & Related papers  (2025-05-22T17:58:12Z)
• PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers [40.097474800631]
  Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Deep neural network based surrogates have gained increased interest.
  arXiv  Detail & Related papers  (2023-08-10T17:53:05Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Neural Stochastic Dual Dynamic Programming [99.80617899593526]
  We introduce a trainable neural model that learns to map problem instances to a piece-wise linear value function. $nu$-SDDP can significantly reduce problem solving cost without sacrificing solution quality.
  arXiv  Detail & Related papers  (2021-12-01T22:55:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.