Bongseok Kim

Amirhossein Mollaali, Bongseok Kim, Christian Moya et al.

Generalizing across disparate physical laws remains a fundamental challenge for artificial intelligence in science. Existing deep-learning solvers are largely confined to single-equation settings, limiting transfer across physical regimes and inference tasks. Here we introduce pADAM, a unified generative framework that learns a shared probabilistic prior across heterogeneous partial differential equation families. Through a learned joint distribution of system states and, where applicable, physical parameters, pADAM supports forward prediction and inverse inference within a single architecture without retraining. Across benchmarks ranging from scalar diffusion to nonlinear Navier--Stokes equations, pADAM achieves accurate inference even under sparse observations. Combined with conformal prediction, it also provides reliable uncertainty quantification with coverage guarantees. In addition, pADAM performs probabilistic model selection from only two sparse snapshots, identifying governing laws through its learned generative representation. These results highlight the potential of generative multi-physics modeling for unified and uncertainty-aware scientific inference.

Bongseok Kim, Jiahao Zhang, Guang Lin

Deep learning has gained attention for solving PDEs, but the black-box nature of neural networks hinders precise enforcement of boundary conditions. To address this, we propose a boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (KAN) with radial basis functions (BEKAN). In BEKAN, we propose three distinct and combinable approaches for incorporating Dirichlet, periodic, and Neumann boundary conditions into the network. For Dirichlet problem, we use smooth and global Gaussian RBFs to construct univariate basis functions for approximating the solution and to encode boundary information at the activation level of the network. To handle periodic problems, we employ a periodic layer constructed from a set of sinusoidal functions to enforce the boundary conditions exactly. For a Neumann problem, we devise a least-squares formulation to guide the parameter evolution toward satisfying the Neumann condition. By virtue of the boundary-embedded RBFs, the periodic layer, and the evolutionary framework, we can perform accurate PDE simulations while rigorously enforcing boundary conditions. For demonstration, we conducted extensive numerical experiments on Dirichlet, Neumann, periodic, and mixed boundary value problems. The results indicate that BEKAN outperforms both multilayer perceptron (MLP) and B-splines KAN in terms of accuracy. In conclusion, the proposed approach enhances the capability of KANs in solving PDE problems while satisfying boundary conditions, thereby facilitating advancements in scientific computing and engineering applications.