Problem-Dependent Artificial Boundary Conditions

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.

In this work, viscous fluid flow governed by the Stokes equations in highly perforated domains is studied using physics-informed neural networks (PINNs). Perforated microstructures induce complex boundary conditions and fine-scale flow features that are difficult for standard neural networks to resolve. Conventional PINNs, even when combined with advanced training techniques, can suffer from a loss of accuracy and efficiency as the number of perforations increases. One important source of this difficulty is the soft enforcement of boundary conditions through penalty terms, which can lead to stiffness, gradient conflicts, and poor resolution of near-boundary flow structures. Hard constraints provide an alternative by encoding boundary conditions exactly into the network ansatz, but may introduce undesirable non-local effects due to the global nature of the approximation. To address these challenges, finite basis PINNs (FBPINNs), which are based on domain decomposition and localisation principles, are used together with hard boundary constraints that efficiently encode perforation-related boundary conditions. This approach helps mitigate spectral bias, improves overall accuracy, and exhibits convergence that is only weakly affected by the number of perforations, thereby providing an efficient and highly parallelisable neural network framework. The proposed approach is further supported with theoretical arguments, specifically focusing on the localisation and approximation properties of FBPINNs.

A flow surrogate validated on a simple regime is often taken as evidence that the approach will carry to a richer one. We test this assumption on two transient flows under time-varying boundary conditions emulating the process startup: the three-dimensional slurry film in chemical-mechanical planarisation (CMP), a core semiconductor-manufacturing process, and the two-dimensional Karman vortex street (KVS) behind a cylinder. Eight surrogate models are compared on one shared evaluation pipeline, differing in whether they learn the full field or a latent representation, and whether they predict trajectories in one shot or step by step. No single architecture wins both regimes. On the film, a one-shot full-field model reconstructs the process-relevant cumulative wall shear stress to 3.2% relative error. On the wake, a latent autoregressive DeepONet retains 96% of the shedding power that direct and one-shot models damp to almost zero. The deciding axis is the treatment of time. The self-sustained wake requires the phase memory that autoregressive feedback provides, while the boundary-driven film rewards a direct map. Pointwise RMSE picks the wrong model in both regimes, so the evaluation scores five physical questions instead, the field, its structure, invented motion, amplitude, and timing. The trained surrogates answer queries $10^3$ to $10^4$ times faster than the finite-element solver, but the offline cost of the training simulations means they pay off from the first query beyond the training set for CMP and the third for the KVS. The choice of surrogate should follow the dynamical character of the target flow, and its validation should use failure-mode-resolved metrics, since neither the winning architecture nor its validation transfers.

Möbius RoPE is a rotary positional encoding built on the anti-periodic frequency ladder $θ_i=π(2i+1)/N$: every rotation plane advances by an odd multiple of $π$ across the training context, so the positional holonomy is $-1$ and the two ends of the sequence are deterministically coupled through a closed-form Dirichlet "dipole"; to our knowledge this is the first anti-periodic boundary condition in positional encoding. We verify the theory numerically to $\sim 10^{-6}$ and pretrain 48 models spanning six 160M-class and three 410M-class arms (2B FineWeb-Edu tokens each; the hybrid arm puts Möbius frequencies on 25% of heads). Hybrid perplexity is unchanged (29.66 vs. 29.72), but needle-in-a-haystack retrieval becomes reliable: $90.3\pm5.7\%$ versus $63.3\pm31.4\%$ at context 512 ($n=6$ seeds), observed worst seed 86% versus 14%, robust variance tests $p=0.013$-$0.029$ (unadjusted), recurring at 410M (Levene $p=0.040$). Matched controls isolate the mechanism: an aperiodic ladder in the same frequency band reproduces none of the effect, and a periodic (holonomy $+1$) ladder only a fraction. Swapping trained models' frequency table back to standard RoPE (weights frozen) collapses retrieval, with damage concentrated on far needles: trained models depend on this long-range geometry. A NoPE arm is even more reliable at short context but pays a 13% perplexity tax and extrapolates worst; only the anti-periodic hybrid pairs baseline perplexity with a high reliability floor. The effect is scoped to single-needle retrieval within the training window; a one-line frequency swap thus provides zero-cost insurance against the retrieval seed lottery.

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov--Galerkin or collocation formulations. We develop a Petrov--Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

Unified visual anomaly detection seeks to train a single detector that can be deployed across categories, domains, and application scenarios. In the few-shot transfer regime, the key challenge is to estimate an episode-specific boundary for an unseen target category from a small support set. Existing approaches mainly infer this boundary from normal-side evidence and provide limited abnormal-side evidence for deployment-specific tolerance. Within the normal side, they often struggle to jointly capture local correspondences and global support-query relations, making their boundaries less reliable for unseen anomalies. To address these issues, we propose UniVAD v2, a two-sided support-conditioned boundary construction framework for unified visual anomaly detection. Built on the component-patch divide-and-conquer framework of UniVAD, UniVAD v2 strengthens the normal side with an Optimal Transport-based Relational Modeling module (OTRM), which complements retrieval with support-query matching through transport-style allocation, and an Adaptive Coordination mechanism for Retrieval and Relational Modeling (ACRRM), which estimates episode-conditioned reliabilities to fuse the two sources of evidence. On the abnormal side, a Few-Shot Abnormal Reference module (FAR) converts optional abnormal references into rejection-side evidence for boundary adjustment. Experiments on six datasets spanning industrial, logical, and medical anomaly detection demonstrate strong cross-domain generalization. Under the 1N-shot protocol, UniVAD v2 improves the mean image-level AUC over UniVAD from 83.0% to 84.5%, and further reaches 85.7% in the 1N+1A-shot setting. On the MVTec-AD Severity Split (MVTec-AD-SS), UniVAD v2 achieves 96.2% image-level AUC and 96.9% pixel-level AUC, showing that abnormal references enable controllable boundary customization without retraining.

Accurate prediction of bottom-hole pressure (BHP) and CO2 plume migration is essential for safe geological carbon storage, yet practical simulations often rely on truncated domains where artificial boundaries distort pressure diffusion and CO2 saturation footprints. In this study, we evaluate how boundary-condition fidelity affects BHP and CO2 plume prediction by comparing ten reduced-domain boundary treatments against full-domain reference simulations in homogeneous and heterogeneous reservoirs. We test uniform pore-volume multipliers, transmissibility modifiers, corner-adjusted pore-volume corrections, layered corrections, and gradual modifiers using BHP RMSE, NRMSE, peak pressure deviation, and plume Intersection over Union (IoU) as performance metrics. Our results show that conserving corner pore volume is the most important requirement for truncated-domain modeling. We find that uniform treatments which neglect corner storage generate large pressure errors, with BHP RMSE of 362 to 382 psi in the homogeneous model and 250 to 304 psi in the heterogeneous model, and yield plume IoU values near 0.80 to 0.84, indicating roughly 16 to 20% of the combined plume area is misrepresented. Corner-adjusted scenarios substantially reduce pressure errors and raise plume IoU above 0.94, but we observe that transmissibility correction is not universally beneficial. In homogeneous reservoirs, uniform transmissibility adjustment improves pressure fidelity; in heterogeneous reservoirs, it can over-restrict flow across variable-permeability boundary faces, increasing BHP error and contracting the predicted plume. We find the gradual modifier with transmissibility correction provides the most consistent performance, achieving BHP NRMSE below 3.7% and plume IoU above 0.97 in both reservoir types.

Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.

We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN). INI-VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks-based formulations, the INI-VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

Aerodynamic simulation is a key component of engineering shape design, where core quantities such as the surface pressure coefficient strongly depend on flow dynamics near solid boundaries. Neural operators provide an efficient alternative to expensive Computational Fluid Dynamics (CFD) solvers. However, conventional methods treat the boundary region isotropically, failing to account for the distinct physical behaviors along the boundaries. In reality, the aerodynamic process exhibits anisotropy: along the tangential direction, flow propagates along the wall; along the normal direction, physical quantities are constrained by the wall. To explicitly model the distinct physical behaviors, we propose GeoABC, a geometry-conditioned anisotropic boundary correction framework. GeoABC leverages the boundary geometries to introduce direction-aware boundary correction into the intermediate representations of neural operators, transforming boundary geometry from static input features into a structural prior that modulates physical prediction. On 2D airfoil and 3D car tasks, GeoABC consistently adapts to multiple neural operator backbones, reducing near-boundary relative $L_2$ error by $\sim$38% on average, narrowing the structural near-wall gap shared by mainstream neural operators, and advancing neural operators toward high-fidelity aerodynamic simulation.

Continual learning struggles to balance retaining past knowledge with absorbing new tasks. Stefan-CL elegantly resolves this stability-plasticity dilemma through the physics of melting. It frames consolidated knowledge as a protected "solid" and unused capacity as an adaptable "liquid." As the network learns, this boundary expands, governed by a "latent heat" tuning dial. By mathematically freezing the learned interior, Stefan-CL cuts forgetting to near zero, matching memory-heavy baselines without storing raw data, forging a beautiful, physics-grounded path for AI.

Single-model ensembles (SMEs) have attracted attention as a way to approximate some of the benefits of deep ensembles within a single network. However, under an approximately matched parameter budget, it remains unclear whether model capacity should be concentrated in a single wide pathway or redistributed into many narrow and independent members. We investigate this question through the Multi-Narrow (MN) transformation, which converts a baseline CNN into an SME of narrow, path-wise independent branches while approximately preserving the dominant parameter budget. We systematically compare Single-Wide and Multi-Narrow configurations across different training-data regimes, architectures, and datasets. The results show that the effectiveness of MN is strongly data-dependent: weakly partitioned or baseline-wide models are preferable in data-rich settings, whereas highly partitioned MN models consistently outperform the baseline in low-data settings. This tendency is reproduced across multiple CNN architectures and image-classification datasets, suggesting that it is not specific to a single benchmark or model family. Analysis of internal representations shows that high-MN models learn more diverse and less redundant path-wise features. In low-data regimes, this diversity is broadly utilized and improves generalization, whereas in data-rich regimes, training becomes imbalanced and prediction is dominated by a small subset of paths. These findings clarify when and why Multi-Narrow transformation is effective, and provide practical guidance for allocating model capacity between width and member multiplicity under a limited budget.

Long-tailed data bias decision boundaries toward head classes and degrade tail class accuracy. Diffusion-based generative augmentation address this problem by generating additional data, while head-to-tail transfer further mitigate the generator bias inherit from long-tailed dataset. However, we show that while head-to-tail transfer helps balance the decision space of the classifier, it also induces latent non-local feature mixing that entangles inter-class features, causing decision boundary overlap and tail class distribution shift. To address this, we first identify the problem of boundary ambiguity and then propose Decision Boundary-aware Generation (DBG) framework, which promotes near-boundary representation learning by generating informative near-boundary samples. Overall, DBG rebalances the long-tailed dataset while yielding more separable decision space for long-tailed learning. Across standard long-tailed benchmarks, DBG consistently improves tail class and overall accuracy with less inter-class overlap. The code of DBG is available at https://github.com/keepdigitalabc-svg/DBG.

3D Gaussian Splatting (3D GS) is widely adopted for novel view synthesis due to its high training and rendering efficiency. However, its efficiency relies on the key assumption that Gaussians do not overlap in the 3D space, which leads to noticeable artifacts and view inconsistencies. In addition, the inherently diffuse boundaries of Gaussians hinder accurate reconstruction of sharp object edges. We propose Softmax-GS, a unified solution that addresses both the view-inconsistency and the diffuse-boundary problem by enforcing a softmax-based competition in overlapping regions between two Gaussians. With learnable parameters controlling the strength of the competition, it enables a continuous spectrum from smooth color blending to crisp, well-defined boundaries. Our formulation explicitly preserves order invariance for any two overlapping Gaussians and ensures that the output transmittance remains unchanged irrespective of the extent of overlapping, preventing undesirable discontinuities in the rendered output. Ablation experiments on simple geometries demonstrate the effectiveness of each component of Softmax-GS, and evaluations on real-world benchmarks show that it achieves state-of-the-art performance, improving both reconstruction quality and parameter efficiency.

With the rapid advancement of intelligent driving and remote sensing, oriented object detection has gained widespread attention. However, achieving high-precision performance is fundamentally constrained by the Angle Boundary Discontinuity (ABD) and Cyclic Ambiguity (CA) problems, which typically cause significant angle fluctuations near periodic boundaries. Although recent studies propose continuous angle coders to alleviate these issues, our theoretical and empirical analyses reveal that state-of-the-art methods still suffer from substantial cyclic errors. We attribute this instability to the structural noise amplification within their non-orthogonal decoding mechanisms. This mathematical vulnerability significantly exacerbates angular deviations, particularly for square-like objects. To resolve this fundamentally, we propose the Fourier Series Coder (FSC), a lightweight plug-and-play component that establishes a continuous, reversible, and mathematically robust angle encoding-decoding paradigm. By rigorously mapping angles onto a minimal orthogonal Fourier basis and explicitly enforcing a geometric manifold constraint, FSC effectively prevents feature modulus collapse. This structurally stabilized representation ensures highly robust phase unwrapping, intrinsically eliminating the need for heuristic truncations while achieving strict boundary continuity and superior noise immunity. Extensive experiments across three large-scale datasets demonstrate that FSC achieves highly competitive overall performance, yielding substantial improvements in high-precision detection. The code will be available at https://github.com/weiminghong/FSC.