computational engineers

Abstract: Operator learning for parametric partial differential equations (PDEs) typically builds global models over prescribed domains, requiring cross-condition data or costly physics-constrained training. Here we show that the output Jacobian of a neural solution model trained at one condition defines a reusable response space for cross-condition solution variations. We introduce Linearized Subspace Transfer (LST) to exploit this space and recover target solutions by minimizing the target PDE-system residual over response-space coordinates. Because any single response space has finite coverage, Active Transfer Modeling (ATM) uses post-transfer residuals as coverage indicators to selectively acquire response spaces from additional single-condition models. Across six systems, single-condition response spaces supported cross-condition transfer, with enrichment improving accuracy when added spaces expanded representation capacity. Relative to evaluated physics-informed operator baselines, ATM reduced error and offline construction cost, with orders-of-magnitude accuracy gains in representative cases and millisecond-to-second target adaptation. These results establish neural solvers as reusable local parametric models.

Abstract: We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

Abstract: Scientific machine learning methods such as physics-informed neural networks (PINNs) increasingly rely on domain decomposition for better scalability while solving partial differential equations (PDEs) over complex geometries, yet the resulting composite loss comprising residual, boundary, and interface terms is highly susceptible to conflicting gradients that degrade training. This work bridges domain decomposition with projection-based gradient surgery to systematically mitigate such conflicts in 2D and 3D settings. We evaluate two existing projection-based algorithms, PCGrad and ConFIG, and identify their performance degradation in specific scenarios such as 3D domains with multiple overlapping interfaces. To address this limitation, we propose Norm-PCGrad, a normalized variant that achieves state-of-the-art accuracy across a range of 2D and 3D domain decomposition problems. Across the benchmarks considered, Norm-PCGrad consistently achieves the lowest relative $L_2$ error compared to training without gradient surgery as well as to existing algorithms such as PCGrad and ConFIG, while incurring negligible additional computational overhead. To improve computational efficiency of domain decomposition frameworks such as Extended PINN (XPINN), we propose replacing vanilla PINNs in selected subdomains with separable architectures such as Separable PINN (SPINN), reducing the computational cost from quadratic (or cubic) to linear. We additionally demonstrate that gradient surgery extends to physics-informed Kolmogorov-Arnold Networks (PIKANs), yielding substantial accuracy improvements for 3D domain decomposition and confirming the generality of the proposed approach across network architectures.

Abstract: Parametric PDEs can admit different boundary conditions with different accuracy and computational cost. We introduce a framework for learning when one reduced boundary condition can replace another: paired solutions train a neural network to estimate the resulting domain and boundary errors, and the simpler condition is used only when both predicted errors meet prescribed tolerances. We focus on singular limits in applications, in which a stiff Robin or nonlinear boundary law is replaced by its limiting Dirichlet form. We evaluate the method on a galvanic corrosion problem and other nonlinear stationary and evolution problems.

Abstract: Localized dimensionality reduction improves the scalability of operator learning for high-dimensional partial differential equations (PDEs), but independently decoded local patches can introduce block offsets, interface mismatches, and spurious high-wavenumber content. We introduce Two-Scale Localized PCA-Net, which decomposes the solution into a coarse-global component and local residual corrections. A compact global PCA basis captures domain-scale structure, while nonoverlapping local PCA bases represent the remaining fine-scale residual. A block-balanced latent objective couples the two representations, and optional interface-aware fine-tuning further promotes continuity through reconstruction and trace losses. On Poisson benchmarks, the two-scale representation substantially reduces reconstruction error and visible block artifacts relative to plain and overlap-based localized PCA-Net while approximately halving PCA fitting cost relative to overlap. On heterogeneous Darcy flow, it strongly reduces interface and discrete-residual errors, with more modest reconstruction gains. Ablations show that the primary improvement arises from the two-scale output representation, while interface-aware fine-tuning provides complementary continuity refinement. Overall, separating globally coherent structure from localized residual detail provides an efficient representation for artifact-reduced PDE operator learning.