Agnostic Neural PDE Solver (MAgNet) is able to make accurate predictions across a variety of PDE simulation datasets and compares favorably with existing baselines.

Motivated by the remarkable success of artificial intelligence (AI) across diverse fields, the application of AI to solve scientific problems-often formulated as partial differential equations (PDEs)-has garnered increasing attention. While most existing research concentrates on theoretical properties (such as well-posedness, regularity, and continuity) of the solutions, alongside direct AI-driven methods for solving PDEs, the challenge of uncovering symbolic relationships within these equations remains largely unexplored. In this paper, we propose leveraging large language models (LLMs) to learn such symbolic relationships. Our results demonstrate that LLMs can effectively predict the operators involved in PDE solutions by utilizing the symbolic information in the PDEs. Furthermore, we show that discovering these symbolic relationships can substantially improve both the efficiency and accuracy of the finite expression method for finding analytical approximation of PDE solutions, delivering a fully interpretable solution pipeline. This work opens new avenues for understanding the symbolic structure of scientific problems and advancing their solution processes.

Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs). However, their generalization capabilities across varying scenarios remain limited. To overcome this limitation, we propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations, including varying initial conditions, PDE coefficients, and training time horizons. PIDO exploits the shared underlying structure of dynamical systems with different properties by projecting PDE solutions into a latent space using auto-decoding. It then learns the dynamics of these latent representations, conditioned on the PDE coefficients. Despite its promise, integrating latent dynamics models within a physics-informed framework poses challenges due to the optimization difficulties associated with physics-informed losses. To address these challenges, we introduce a novel approach that diagnoses and mitigates these issues within the latent space. This strategy employs straightforward yet effective regularization techniques, enhancing both the temporal extrapolation performance and the training stability of PIDO. We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations. Additionally, we demonstrate the transferability of its learned representations to downstream applications such as long-term integration and inverse problems.

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which learn mappings from parameters to solutions, have been successfully used. However, the training of neural operators typically demands large training datasets, the acquisition of which can be prohibitively expensive. To address this challenge, physics-informed training can offer a cost-effective strategy. However, current physics-informed neural operators face limitations, either in handling irregular domain shapes or in generalization to various discretizations of PDE parameters with variable mesh sizes. In this research, we introduce a novel physics-informed model architecture which can generalize to parameter discretizations of variable size and irregular domain shapes. Particularly, inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly, and this parameter embedding is integrated with the response embeddings through multiple compositional layers, for more expressivity. Numerical results demonstrate the accuracy and efficiency of the proposed method.

Physics-Informed Neural Networks (PINNs), which incorporate PDEs as soft constraints, train with a composite loss function that contains multiple training point types: different types of collocation points chosen during training to enforce each PDE and initial/boundary conditions, and experimental points which are usually costly to obtain via experiments or simulations. Training PINNs using this loss function is challenging as it typically requires selecting large numbers of points of different types, each with different training dynamics. PINNACLE uses information on the interaction among training point types, which had not been considered before, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). We theoretically show that the criterion used by PINNACLE is related to the PINN generalization error, and empirically demonstrate that PINNACLE is able to outperform existing point selection methods for forward, inverse, and transfer learning problems. Deep learning (DL) successes in domains with massive datasets have led to questions on whether it can also be efficiently applied to the scientific domains. In these settings, while training data may be more limited, domain knowledge could compensate by serving as inductive biases for DL training. Such knowledge can take the form of governing Partial Differential Equations (PDEs), which can describe phenomena such as conservation laws or dynamic system evolution in areas such as fluid dynamics (Cai et al., 2021; Chen et al., 2021; Jagtap et al., 2022), wave propagation and optics (bin Waheed et al., 2021; Lin & Chen, 2022), or epidemiology (Rodríguez et al., 2023). Physics-Informed Neural Networks (PINNs) are neural networks that incorporate PDEs and their initial/boundary conditions (IC/BCs) as soft constraints during training (Raissi et al., 2019), and have been successfully applied to various problems. These include forward problems (i.e., predicting PDE solutions given specified PDEs and ICs/BCs) and inverse problems (i.e., learning unknown PDE parameters given experimental data). However, the training of PINNs is challenging. Past works have tried to separately address these individually by considering an adaptive selection of CL points (Nabian et al., 2021; Gao & Wang, 2023; Wu et al., 2023; Peng et al., 2022; Zeng et al., 2022; Tang et al., 2023), or E Some works have also proposed heuristics that adjust loss term weights of the various point types to try improve training dynamics, but do not consider point selection (Wang et al., 2022c). However, no work thus far has looked into optimizing all training point types jointly to significantly boost PINN performance. Given that the solution spaces of the PDE, IC/BC and underlying output function are tightly coupled, it is inefficient and sub-optimal to select each type of training points separately and ignore cross information across point types.

Deep neural networks (DNNs) are increasingly used to solve partial differential equations (PDEs) that naturally arise while modeling a wide range of systems and physical phenomena. However, the accuracy of such DNNs decreases as the PDE complexity increases and they also suffer from spectral bias as they tend to learn the low-frequency solution characteristics. To address these issues, we introduce Parametric Grid Convolutional Attention Networks (PGCANs) that can solve PDE systems without leveraging any labeled data in the domain. The main idea of PGCAN is to parameterize the input space with a grid-based encoder whose parameters are connected to the output via a DNN decoder that leverages attention to prioritize feature training. Our encoder provides a localized learning ability and uses convolution layers to avoid overfitting and improve information propagation rate from the boundaries to the interior of the domain. We test the performance of PGCAN on a wide range of PDE systems and show that it effectively addresses spectral bias and provides more accurate solutions compared to competing methods.

The solution of a PDE over varying initial/boundary conditions on multiple domains is needed in a wide variety of applications, but it is computationally expensive if the solution is computed de novo whenever the initial/boundary conditions of the domain change. We introduce a general operator learning framework, called DIffeomorphic Mapping Operator learNing (DIMON) to learn approximate PDE solutions over a family of domains $\{\Omega_{\theta}}_\theta$, that learns the map from initial/boundary conditions and domain $\Omega_\theta$ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain $\Omega_{\theta}$) to a problem on a reference domain $\Omega_{0}$, where training data from multiple problems is used to learn the map to the solution on $\Omega_{0}$, which is then re-mapped to the original domain $\Omega_{\theta}$. We consider several problems to demonstrate the performance of the framework in learning both static and time-dependent PDEs on non-rigid geometries; these include solving the Laplace equation, reaction-diffusion equations, and a multiscale PDE that characterizes the electrical propagation on the left ventricle. This work paves the way toward the fast prediction of PDE solutions on a family of domains and the application of neural operators in engineering and precision medicine.