Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers

• URL: http://arxiv.org/abs/2606.22150v1
• Date: Sat, 20 Jun 2026 17:02:30 GMT
• Title: Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers
• Abstract summary: Implicit Modulation (PRISM) is a plug-and-play framework for parameterized high-dimensional and high-order neural PDE solvers.<n>We show that it entangles physical parameters with the high-order automatic differentiation, causing extra memory growth and parameter variance amplification.<n>Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU.
• Score: 0.24554686192257416
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.

Related papers

• Equation Recast for Canonical Operator Learning Across Parametric PDEs [59.44573033543284]
  We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator.<n>In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries.
  arXiv  Detail & Related papers  (2026-09-02T13:15:50Z)
• Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers [0.24554686192257416]
  Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy.<n>We introduce EMNGDfull, a manifold optimization framework for physics-informed and variational neural PDE solvers.
  arXiv  Detail & Related papers  (2026-07-24T06:07:14Z)
• AE-ViT: Stable Long-Horizon Parametric Partial Differential Equations Modeling [0.0]
  We propose a joint model consisting of a convolutional encoder, a transformer operating on latent representations, and a decoder for reconstruction.<n>Our approach combines the efficiency of latent evolution with the fidelity of full-field models, outperforming DL-ROMs, latent transformers, and plain ViTs in multi-field prediction.
  arXiv  Detail & Related papers  (2026-04-07T21:19:45Z)
• Generalized Tensor-based Parameter-Efficient Fine-Tuning via Lie Group Transformations [50.010924231754856]
  Adapting pre-trained foundation models for diverse downstream tasks is a core practice in artificial intelligence.<n>To overcome this, parameter-efficient fine-tuning (PEFT) methods like LoRA have emerged and are becoming a growing research focus.<n>We propose a generalization that extends matrix-based PEFT methods to higher-dimensional parameter spaces without compromising their structural properties.
  arXiv  Detail & Related papers  (2025-04-01T14:36:45Z)
• GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning [14.939978372699084]
  Data-driven approaches learn parametric PDEs by incorporating a very large variety of trajectories with varying PDE parameters. GEPS is a simple adaptation mechanism to boost GEneralization in Pde solvers. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers.
  arXiv  Detail & Related papers  (2024-10-31T12:51:40Z)
• Adapting Projection-Based Reduced-Order Models using Projected Gaussian Process [5.492716202049269]
  We propose a Projected Gaussian Process (pGP) to learn a mapping from the parameter space to the Grassmann manifold that contains the optimal subspaces.<n>As a statistical learning approach, the proposed pGP allows us to optimally estimate (or tune) the model parameters from data and quantify the statistical uncertainty associated with the prediction.
  arXiv  Detail & Related papers  (2024-10-18T00:02:43Z)
• Scaling Exponents Across Parameterizations and Optimizers [94.54718325264218]
  We propose a new perspective on parameterization by investigating a key assumption in prior work. Our empirical investigation includes tens of thousands of models trained with all combinations of threes. We find that the best learning rate scaling prescription would often have been excluded by the assumptions in prior work.
  arXiv  Detail & Related papers  (2024-07-08T12:32:51Z)
• Maintaining Structural Integrity in Parameter Spaces for Parameter Efficient Fine-tuning [78.39310274926535]
  Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence.<n>To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner.<n>This paper introduces a generalized parameter-efficient fine-tuning framework, designed for various dimensional parameter space.
  arXiv  Detail & Related papers  (2024-05-23T16:04:42Z)
• Optimization of Annealed Importance Sampling Hyperparameters [77.34726150561087]
  Annealed Importance Sampling (AIS) is a popular algorithm used to estimates the intractable marginal likelihood of deep generative models. We present a parameteric AIS process with flexible intermediary distributions and optimize the bridging distributions to use fewer number of steps for sampling. We assess the performance of our optimized AIS for marginal likelihood estimation of deep generative models and compare it to other estimators.
  arXiv  Detail & Related papers  (2022-09-27T07:58:25Z)
• Sliced gradient-enhanced Kriging for high-dimensional function approximation [2.8228516010000617]
  Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. It tends to get impractical for high-dimensional problems due to the size of the inherent correlation matrix. A new method, called sliced GE-Kriging (SGE-Kriging), is developed in this paper for reducing the size of the correlation matrix. The results show that the SGE-Kriging model features an accuracy and robustness that is comparable to the standard one but comes at much less training costs.
  arXiv  Detail & Related papers  (2022-04-05T07:27:14Z)
• Implicit differentiation of Lasso-type models for hyperparameter optimization [82.73138686390514]
  We introduce an efficient implicit differentiation algorithm, without matrix inversion, tailored for Lasso-type problems. Our approach scales to high-dimensional data by leveraging the sparsity of the solutions.
  arXiv  Detail & Related papers  (2020-02-20T18:43:42Z)

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.