A physics-aware neural-network-based latent-space framework for reduced-order forward modeling and variational parameter estimation and the robustness of the proposed framework is demonstrated, which generally reduces calibration error and variability compared with the standard surrogate models.
Abstract
Forward and inverse modeling of parametric dynamical systems requires surrogate models that are not only accurate for state prediction, but also informative for parameter calibration. However, a systematic end-to-end differentiable formulation for coupling deep-learning-based reduced-order surrogates with variational parameter estimation remains underdeveloped. In this work, we introduce a physics-aware neural-network-based latent-space framework for reduced-order forward modeling and variational parameter estimation. The proposed autoencoder-based approach yields a differentiable surrogate that maps physical parameters to predicted flow fields through a latent representation. The observable supervision is used during offline training to encourage the latent variables to retain information correlated with system parameters, while the online inverse problem is solved in the parameter space through the surrogate-induced observation operator. The method is evaluated on two computational-fluid-dynamics benchmarks. The results show that reconstruction accuracy alone is insufficient for inverse modeling, owing to the lack of end-to-end differentiability or physics awareness for variational parameter calibration. Quantitative latent-space analysis further shows that observable supervision improves case-level separability and temporal organization of latent representations. Experiments with realistic measurement settings, including noisy, low-resolution, randomly masked, and block-wise partial observations, demonstrate the robustness of the proposed framework and show that it generally reduces calibration error and variability compared with the standard surrogate models.
Nonlinear state-space model identification is inherently challenging due to the need for joint latent-state inference and parameter learning. Variational inference offers a tractable framework, where structured Gaussian posteriors enable scalable inference over latent trajectories. However, existing parameterizations of the posterior mean struggle to capture complex nonlinear dynamics, while more expressive deep parameterizations tend to introduce instability in joint optimization. To address these issues, this article proposes a deep variational identification method for nonlinear state-space models based on a structured Gaussian posterior. Specifically, the posterior mean is parameterized using a non-causal dilated residual convolutional network, while a Markov structure with block-tridiagonal precision is preserved to ensure linear-time inference. Furthermore, an alternating optimization scheme is developed to separate variational smoothing from model identification. The variational parameters are updated by maximizing a differentiable approximation to the evidence lower bound, whereas the model and noise parameters are updated via analytic identification steps using samples from the variational posterior. Experiments on a nonlinear discrete-time system and a stochastic Duffing oscillator demonstrate that the proposed approach achieves stable optimization and reliable estimation of the system dynamics and noise statistics.
Kaiqi Fang, Guijun Ma, Yasen Wang et al.· International Journal of Net...· 0 citations
Reduced-order models, such as latent dynamics models, are becoming mainstream for accelerating simulations for parameterized physical systems governed by nonlinear conservation laws. However, most existing latent dynamics frameworks suffer from two important limitations: they do not provide uncertainty estimates for model predictions, and they do not guarantee adherence to the underlying conservation laws. While these challenges have been addressed separately in prior work, a unified framework that simultaneously provides uncertainty quantification and exact conservation-law preservation remains largely unexplored. In this work, we develop a variational latent neural field framework that integrates Gaussian process-inspired surrogates, enabling estimation of predictive confidence for both in-distribution and out-of-distribution parameter regimes. Three variants of the framework are considered: IRS-UQ, PI-IRS-UQ, and ECLEIRS-UQ, corresponding to unconstrained, physics-informed, and conservation-structure-preserving formulations, respectively. Exact conservation-structure preservation is achieved by embedding the solution dynamics within a conservation-law manifold through a space-time divergence-free representation of the solution-flux field. We demonstrate the applicability of the framework through three numerical experiments: 1) 1-D advection, 2) 2-D Euler and 3) 2-D shallow water equations in parameterized settings. Numerical experiments demonstrate that the proposed approach provides accurate predictions together with uncertainty estimates, while remaining robust to sparse and noisy training data. Comparisons between the proposed three approaches show that conservation-structure preserving latent representations improve robustness to degraded training data while maintaining competitive predictive accuracy and uncertainty quantification capability.
Deep learning has proven highly effective for nonlinear system identification, but heavily parameterized neural networks are prone to overfitting in low-data regimes and lack reliable uncertainty quantification. The recently developed manifold meta-learning framework addresses the data efficiency problem by restricting the model parameters to a meta-learned low-dimensional manifold. However, that method is purely deterministic. We propose a fully probabilistic extension of the manifold meta-learning framework, based on amortized Variational Inference, where a generative prior over the low-dimensional parameter manifold is learned. During task-specific adaptation, we combine Maximum A Posteriori estimation with the Laplace approximation to yield a mathematically grounded posterior approximation. Evaluated on a static regression task and the Bouc--Wen dynamical system benchmark, the proposed approach achieves predictive accuracy comparable to its deterministic counterpart while successfully providing calibrated uncertainty bounds in severely low-data regimes.
A Fourier-enhanced operator autoencoder for decoder-free reconstruction and latent learning of dynamical systems and achieves accuracy comparable to or better than classical AE-based reduced-order models while providing a more efficient latent-to-field reconstruction path.
Xuandong Lu, Yongming Liu· Machine Learning for Computa...· 0 citations
Physics-informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems. However, their direct application to structural dynamics remains limited by high system dimensionality and model-form errors arising from incomplete physics. Reduced-order models (ROMs) can alleviate the dimensionality bottleneck, yet existing PINN-ROM couplings typically rely on fixed reduced subspaces, target forward simulations, or assume complete physics, restricting their use for inverse identification under parametric variability or incomplete system knowledge. To address these limitations, this work proposes a Reduced-Order Physics-Informed Neural Network (RO-PINN) framework with adaptive basis refinement for structural identification under known and incomplete physics. Via projection, reduced governing equations are embedded directly into the PINN loss, facilitating learning in a low-dimensional latent space. An adaptive scheme updates the projection basis during training so that the latent space is progressively realigned with evolving structural parameters or learned residual restoring forces. This realignment reduces basis-mismatch errors and limits their influence on the inferred residual force. The method is validated on a four-story steel frame with nonlinear hysteretic braces under sparse and noisy measurements. Results show parameter identification comparable to or more accurate than Bayesian model updating with lower computational cost in the considered cases, recovery of unmodeled nonlinear restoring forces under incomplete physics, and joint identification of residual restoring forces and structural parameters within the same framework. Overall, RO-PINN provides a unified framework for structural identification by integrating reduced-order modeling, adaptive basis refinement, and physics-informed learning within a single formulation.
Physics-informed machine learning of parametric partial differential equation (PDE) families enables rapid prediction across varying physical conditions, yet the resulting task representations are commonly embedded in latent neural features that are difficult to interpret physically. This raises the question of whether a parametric neural PDE solver can make explicit how physical task parameters reorganize the underlying solution representation. To address this gap, we introduce Meta-Sparse, Physics-based, and partially Interpretable Neural Network (SPINN), which maps task parameters to a shallow RBF model with inspectable coefficients, centers, scales, and directional parameters. We show that, across elliptic, transport, advection–diffusion, variable-coefficient, and nonlinear PDE families, the learned bases adapt to and organize around the dominant physical solution structures, including localized forcing responses, characteristic-aligned transport trajectories, diffusion-broadened space–time corridors, and viscous shock fronts. Meta-SPINN works both as a direct predictor for unseen tasks and as a task-aware initializer for subsequent single-instance residual-guided refinement, providing reusable predictions together with an interpretable visualization of how solution geometry changes across a parameter family.