We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers.

PINNs are, however, sharing the same weakness with coordinate-based MLPs (or INRs), which hinders the application of PINNs/INRs to more diverse applications; for a new data instance (e.g., a new PDE for PINNs or a new image for INRs), training a new neural network (typically from

Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle to capture this variability. To address this, data-driven approaches learn parametric PDEs by sampling a very large variety of trajectories with varying PDE parameters. We first show that incorporating conditioning mechanisms for learning parametric PDEs is essential and that among them, \textit{adaptive conditioning}, allows stronger generalization. As existing adaptive conditioning methods do not scale well with respect to the number of parameters to adapt in the neural solver, we propose GEPS, a simple adaptation mechanism to boost GEneralization in Pde Solvers via a first-order optimization and low-rank rapid adaptation of a small set of context parameters. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers. Validation performed on a whole range of spatio-temporal forecasting problems demonstrates excellent performance for generalizing to unseen conditions including initial conditions, PDE coefficients, forcing terms and solution domain.

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc. Recently, building learning-based numerical solvers for parametric PDEs has become an emerging new field.

We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers.

PINNs are, however, sharing the same weakness with coordinate-based MLPs (or INRs), which hinders the application of PINNs/INRs to more diverse applications; for a new data instance (e.g., a new PDE for PINNs or a new image for INRs), training a new neural network (typically from

Inferring parameters of high-dimensional partial differential equations (PDEs) poses significant computational and inferential challenges, primarily due to the curse of dimensionality and the inherent limitations of traditional numerical methods. This paper introduces a novel two-stage Bayesian framework that synergistically integrates training, physics-based deep kernel learning (DKL) with Hamiltonian Monte Carlo (HMC) to robustly infer unknown PDE parameters and quantify their uncertainties from sparse, exact observations. The first stage leverages physics-based DKL to train a surrogate model, which jointly yields an optimized neural network feature extractor and robust initial estimates for the PDE parameters. In the second stage, with the neural network weights fixed, HMC is employed within a full Bayesian framework to efficiently sample the joint posterior distribution of the kernel hyperparameters and the PDE parameters. Numerical experiments on canonical and high-dimensional inverse PDE problems demonstrate that our framework accurately estimates parameters, provides reliable uncertainty estimates, and effectively addresses challenges of data sparsity and model complexity, offering a robust and scalable tool for diverse scientific and engineering applications.

Below is a detailed explanation of the comparative methods covered in the paper. The network architecture of PI-DeepONet used for Burgers' equation is such that both In order to solve the Eq. Fig.6 shows model predictions of MAD-L and MAD-LM compared with the reference solutions under Fig.7(a) shows that the accuracy of MAD-L after convergence increases with Fig.7(b) shows that the accuracy and convergence speed of MAD-LM do not change For Burgers' equation, we also consider the scenario when the viscosity coefficients Fig.8 compares the convergence curves of mean MAD-LM has obvious speed and accuracy improvement over From-Scratch and Transfer-Learning . We investigated the effect of the dimension of the latent vector (latent size) in Burgers' equation on performance. As can be seen from Fig.9(a), for MAD-L, different latent sizes have different performances and the best performance is achieved when it is equal to 128.