Empirical Hölder-regularized PINN objective

Inputs are a dimension $d\ge 1$, the data and operators specified in the physics-informed residual loss definition, $0<\alpha \le 1$, positive weights ${\lambda }_r,{\lambda }_b$, and an admissible trial function $h$. The procedure returns the scalar empirical objective value $J_m(h)$. It uses the realized sample points and target values as inputs; it does not generate random samples.

For $d\ge 2$, also supply positive integer counts $m_r,m_b$, residual samples $x_r^1,\dots ,x_r^{m_r}\in U$ with their observed values $f(x_r^i)$, boundary samples $x_b^1,\dots ,x_b^{m_b}\in \mathrm{\Gamma }$ with their observed values $g(x_b^j)$, constants $c_r,c_b,C_r,C_b>0$, and small-scale cutoffs ${\mathrm{\ell }}_{0,i},{\rho }_{0,i}>0$. At the selected partition scales $0<{\mathrm{\ell }}_i\le {\mathrm{\ell }}_{0,i}$, require cells $U_j^{{\mathrm{\ell }}_r}$ and ${\mathrm{\Gamma }}_j^{{\mathrm{\ell }}_b}$ with masses at least $c_r{\mathrm{\ell }}_r^d$ and $c_b{\mathrm{\ell }}_b^{d-1}$, each contained in a cube of side ${\mathrm{\ell }}_i$. Require upper ball-mass bounds ${\mu }_r({\bar{B}}_{\rho }(x)\cap U)\le C_r{\rho }^d$ and ${\mu }_b({\bar{B}}_{\rho }(y)\cap \mathrm{\Gamma })\le C_b{\rho }^{d-1}$ for every center and for all $0<\rho \le {\rho }_{0,i}$. The admissible counts satisfy, for $i\in \{r,b\}$, $m_i\ge \max \{1,c_i^{-2}{\mathrm{\ell }}_{0,i}^{-2s_i},c_i^{-2}{d}^{s_i}{\rho }_{0,i}^{-2s_i}\},$ so the computed partition scales and fill radii lie in the specified ranges.

For $d=1$, supply a positive integer $m_r$, residual samples $x_r^1,\dots ,x_r^{m_r}\in U$ with observed values $f(x_r^i)$, constants $c_r,C_r>0$, and cutoffs ${\mathrm{\ell }}_{0,r},{\rho }_{0,r}>0$. At the residual scale ${\mathrm{\ell }}_r$ below, require a finite measurable partition with each cell contained in an interval of length ${\mathrm{\ell }}_r$ and mass at least $c_r{\mathrm{\ell }}_r$, and require upper ball-mass bounds ${\mu }_r({\bar{B}}_{\rho }(x)\cap U)\le C_r\rho$ for every center $x\in U$ and all $0<\rho \le {\rho }_{0,r}$. The admissible count satisfies $m_r\ge \max \{1,c_r^{-2}{\mathrm{\ell }}_{0,r}^{-2},c_r^{-2}{\rho }_{0,r}^{-2}\},$ so ${\mathrm{\ell }}_r={\delta }_r$ is in the specified ranges. Supply the observed boundary values at every point of the finite set $\mathrm{\Gamma }$ with their masses ${\mu }_b(\{y\})$. No random boundary sample or boundary sample count is used.

For $d\ge 2$, set $s_r=d$, $s_b=d-1$, and, for $i\in \{r,b\}$, compute ${\mathrm{\ell }}_i=(c_i\sqrt{m_i}{)}^{-1\mathrm{/}{s}_i},{\delta }_i=\sqrt{d}{\mathrm{\ell }}_i,{\kappa }_i=C_i\mathrm{/}{c}_i,$ $C_m=3{\max }_{i\in \{r,b\}}\{{\kappa }_i{d}^{s_i\mathrm{/}2}{m}_i^{1\mathrm{/}2}\},{\widehat{\lambda }}_{i,m}^R=\frac{3{\lambda }_i{\delta }_i^{2\alpha }}{C_m}\mathrm{.}$ For $d=1$, compute ${\mathrm{\ell }}_r={\delta }_r=(c_r\sqrt{m_r}{)}^{-1},{\kappa }_r=C_r\mathrm{/}{c}_r,C_m=3{\kappa }_r{m}_r^{1\mathrm{/}2},$ ${\widehat{\lambda }}_{r,m}^R=\frac{3{\lambda }_r{\delta }_r^{2\alpha }}{C_m}=\frac{{\lambda }_r{c}_r^{-2\alpha }}{{\kappa }_r}{m}_r^{-\alpha -1\mathrm{/}2},{\widehat{\lambda }}_{b,m}^R=0.$ For $d\ge 2$, use supplied regularization weights ${\lambda }_{i,m}^R$ satisfying ${\lambda }_{i,m}^R\ge {\widehat{\lambda }}_{i,m}^R$ componentwise; for a sequence of sample counts also require $\mathrm{\parallel }{\lambda }_m^R{\mathrm{\parallel }}_{\mathrm{\infty }}=O(\mathrm{\parallel }{\widehat{\lambda }}_m^R{\mathrm{\parallel }}_{\mathrm{\infty }})$. For $d=1$, set ${\lambda }_{b,m}^R=0$ and choose ${\lambda }_{r,m}^R\ge {\widehat{\lambda }}_{r,m}^R$, with the corresponding one-coordinate max-norm order condition across a sequence of counts. If no residual weights are supplied, set them equal to their thresholds.

Finally evaluate and return $J_m(h)={\mathrm{Loss}}_m(h;\lambda ,{\lambda }_m^R)$ using the realized residual values, the realized boundary values when $d\ge 2$, the exact weighted full-boundary contribution when $d=1$, and the residual seminorm penalty when $d=1$ or both seminorm penalties when $d\ge 2$. The regularization thresholds are the source's exact thresholds and cannot be replaced by a generic positive penalty.

physics informed neural network loss

Shin et al., equations (3.1)–(3.3): exact thresholds, scaling condition, and objective

For $d\ge 2$, the exact residual and boundary threshold coordinates are ${\widehat{\lambda }}_{r,m}^R=\frac{3{\lambda }_r(\sqrt{d}{)}^{2\alpha }{c}_r^{-2\alpha \mathrm{/}d}{m}_r^{-\alpha \mathrm{/}d}}{C_m},{\widehat{\lambda }}_{b,m}^R=\frac{3{\lambda }_b(\sqrt{d}{)}^{2\alpha }{c}_b^{-2\alpha \mathrm{/}(d-1)}{m}_b^{-\alpha \mathrm{/}(d-1)}}{C_m},$ where $C_m=3\max \{(C_r\mathrm{/}{c}_r)(\sqrt{d}{)}^d{m}_r^{1\mathrm{/}2},(C_b\mathrm{/}{c}_b)(\sqrt{d}{)}^{d-1}{m}_b^{1\mathrm{/}2}\}$. For $d=1$, the residual threshold is ${\widehat{\lambda }}_{r,m}^R=({\lambda }_r{c}_r^{-2\alpha }\mathrm{/}{\kappa }_r)m_r^{-\alpha -1\mathrm{/}2}$, with ${\kappa }_r=C_r\mathrm{/}{c}_r$, and the boundary threshold is zero. The weight vector must satisfy the componentwise lower bound and max-norm order condition in the Algorithm.

The original all-$\epsilon >0$ sampling quantifier has no probability-measure instances; it is preserved and refuted in the all-scale inconsistency lemma. The small-scale range above is the corrected active sampling convention. No lower-mass constant normalization is implicit in this algorithm.