Replace or augment relative-position attention with a positive fractional-integration mixing kernel whose radial behavior has separate inner and outer power laws. Tokens close to one another interact through the usual fractional singularity, while tokens near different radial scales receive a ground-state correction that can improve multiscale information transport without introducing a dense learned positional table.
Train a value network for stopping or intervention decisions using a killed-resolvent identity rather than an unrestricted diffusion residual. Simulating only until the process exits the continuation region makes the learning target local to the relevant decision domain and correctly handles nonsmooth max rewards.
Add a topological loss that preserves the winding number of a complex numerator field predicted by a neural network. The loss is invariant to positive rescaling of the field, so it penalizes vortex creation or destruction rather than harmless amplitude changes.
Add a two-channel geometric head producing scalar fields u(x) and v(x) on a two-dimensional input or latent coordinate domain. Train it initially with a moderate p-harmonic duality constraint, then anneal p upward so u approaches an infinity-harmonic field while v remains its rotated-gradient dual; this penalizes isolated steep gradient spikes and promotes smooth, coherent level sets.
Apply the paper's reversed weighted interaction inequality to two nonnegative feature maps generated from different augmentations or network branches. Maximizing the normalized nonlocal interaction should discourage collapsed or overly concentrated spatial representations while remaining invariant to overall feature amplitude.
Split the trainable state into an explicit scalar scale coordinate and a residual perturbation, then update them with separate time scales. Penalize residuals according to their distance from the scale-dependent core, so the optimizer cannot obtain apparent progress by destabilizing the scale mode. The method is a neural optimization analogue of the paper's modulation argument, not a direct consequence of the geometric singularity theorem.
Represent the prediction as a sum of a smooth interior branch and a fractional boundary branch: u_theta(x)=u_int_theta(x)+d(x)^a u_bd_theta(x). This mirrors the paper's direct-sum solution structure and allocates separate network capacity to the globally regular component and the boundary layer.
Build a decoder \(F:\mathbb{R}^m\to\mathbb{R}^N\) whose latent-coordinate derivatives are approximately horizontal, meaning they annihilate a prescribed one-form \(\lambda\). When \(\lambda\wedge d\lambda=0\), use local chart-wise training or Jacobian projection to exploit the paper's Lipschitz extension regime and obtain smoother, geometrically valid interpolations between observed boundary samples.
Regularize learned low-dimensional embeddings or MoE prototypes with an aggregation-diffusion energy. The attractive term encourages compact, semantically coherent groups, while porous-medium diffusion creates density-dependent pressure that prevents points from collapsing into singular clusters.
Add a fixed or weakly learned interface-localized branch to a sequence model. Set the critical mass term to zero and make the transport coefficient change sign across a learnable interface, producing a localized mode that pools information near a detected transition rather than averaging uniformly over the sequence.
Augment spatial training examples by replacing a compact active region with several separated components while preserving its exact value histogram, total active area, and amplitude. The augmentation probes the nonlinear interaction between diffusion-like receptive fields and threshold activations, which the paper shows can make fragmented and compact inputs evolve in opposite directions despite identical distributions.