Beyond Residuals: Guaranteeing Accuracy in Neural PDE Solutions

A new framework provides verifiable bounds on the error of neural network-based solvers for partial differential equations, moving beyond empirical observation to solution-space guarantees.
![QuaQue facilitates translation from [latex]\text{SPARQL}[/latex] to [latex]\text{SQL}[/latex], enabling queries across knowledge graphs and relational databases through a unified interface.](https://arxiv.org/html/2603.18654v1/assets/traductions-de-modeles.png)
![The study presents continuum-extrapolated results for tetraquark states-specifically, the axialvector and scalar [latex]b\bar{s}\bar{u}\bar{d}[/latex]-normalized against their respective decay thresholds of [latex]B^*K[/latex] and [latex]BK[/latex], with uncertainties quantified by [latex]1\sigma[/latex] bands, offering insights into the composition and stability of these exotic hadronic structures.](https://arxiv.org/html/2603.18667v1/x8.png)

![Predictions within the Standard Model detail the differential branching fraction and angular observables-specifically [latex]F_L^L[/latex] and [latex]A_{FB}^\ell[/latex]-resulting from the decay of the [latex]\Xi_b^-\to\Xi^-\mu^+\mu^-[/latex] particle.](https://arxiv.org/html/2603.18438v1/x21.png)
![Distributions of neuron activations at bottleneck layers and the fully connected layer reveal that [latex]\mathcal{M}[/latex] (blue) and [latex]\widetilde{\mathcal{M}}[/latex] (orange) models exhibit distinct probability densities during path selection, as indicated by quartile markers and mean values.](https://arxiv.org/html/2603.19025v1/x1.png)

![Tetraquark operators are constructed from displaced quarks and antiquarks arranged in single-site, I-shaped doubly-displaced, and cross-shaped quadruply-displaced configurations, with variations denoted by ‘a’ and ‘b’ representing differing displacement orderings and encompassing two distinct color structures as defined by [latex] Eqs. (12) [/latex] and [latex] (13) [/latex].](https://arxiv.org/html/2603.19192v1/x1.png)