Papers

We introduce a framework for retraction-based convex optimization on Riemannian manifolds, which includes a notion of retraction-specific convex sets and retraction-based gradient projection algorithms. The standard theory of gradient projection algorithms generalizes easily to this framework. Within this framework, we establish convergence results for retraction-based gradient projection algorithms with various stepsize rules. As an application, we use our framework to study the weighted low-rank approximation. We also provide numerical validation of our convergence results on the image completion task.

We study whether the linear condition-number dependence in the stochastic complexity of SAPD+ is necessary for nonconvex-strongly-concave minimax optimization. For jointly $L$-smooth objectives with dual strong-concavity parameter $\mu$, we prove a lower bound that matches the SAPD+ upper bound under the same Moreau-envelope stationarity criterion and the same primal-dual initialization gap. Specifically, when $\sigma\ge\varepsilon$, the worst-case complexity of zero-respecting algorithms is $\Theta(\kappa LG\sigma^2\varepsilon^{-4})$ in the stated accuracy regime, where $\kappa=L/\mu$, $G$ bounds the initial primal-dual gap, and $\sigma^2$ bounds the variance of a general unbiased first-order oracle. The lower bound is realized on a smooth problem class with a bounded dual box. Our construction routes each link of a nonconvex zero-chain through a dual gradient of magnitude proportional to $\varepsilon/\sqrt{\kappa}$, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound $\Omega(L\Delta(\sqrt{\kappa}\varepsilon^{-2}+\kappa\sigma^2\varepsilon^{-4}))$ after combination with the known deterministic bound, where $\Delta$ bounds the initial primal function gap.

We propose a vanishing-viscosity minimizing movement (VVMM) Fourier spectral method for scalar hyperbolic conservation laws under periodic boundary conditions. The method is a fully discrete space–time residual-minimizing scheme, Fourier spectral in space and polynomial in time, stabilized by vanishing viscosity and heat-kernel postprocessing. A basic analytical difficulty in Fourier discretizations of nonlinear conservation laws is the interaction between Fourier projection and the nonlinear flux; the resulting projection residual contains unresolved high-frequency components. VVMM addresses this issue by selecting, on each time slab, the numerical solution that minimizes {a penalized cut-off viscous conservation laws residual} and then applies heat-kernel postprocessing to damp unresolved modes. We prove a fully discrete $L^1$ error estimate, in any fixed space dimension, that separates the vanishing-viscosity error, the temporal interpolation error, and the spatial Fourier truncation error. {Under exact minimization of the penalized cut-off residual and suitable parameter coupling,} the estimate yields, for each fixed $\gamma>0$, an $L^1$ convergence rate $N^{-1/2+\gamma}$, with a constant depending on $\gamma$. Numerical experiments in one and two dimensions illustrate stable shock capturing without visible Gibbs oscillations and convergence behavior consistent with the theory.

We compute the rational homology of $\mathrm{Out}(F_8)$ and $\mathrm{Aut}(F_7)$, and several related groups. We also show that the Eisenstein classes in $H_7(\mathrm{Aut}(F_5))$ and $H_{11}(\mathrm{Aut}(F_7))$ are zero.

Let $\theta\in\mathbb Q\setminus\{0\}$. We construct an approximating function $\psi:\mathbb N\to[0,\frac12)$ for which \[ \sum_{q=1}^{\infty}\frac{\varphi(q)}{q}\psi(q)=\infty, \] but the set of $x\in[0,1]$ for which \[ |qx-a-\theta|<\psi(q),\qquad \gcd(a,q)=1, \] holds for infinitely many $(a,q)\in\mathbb Z\times\mathbb N$ has Lebesgue measure zero. Thus the inhomogeneous analogue of the Duffin–Schaeffer conjecture fails for every nonzero rational shift. By a modification of the construction, we further show that the set of shifts for which the inhomogeneous Duffin–Schaeffer conjecture fails is residual in $\mathbb R$.

Quantum-inspired evolutionary optimization (QIEO) is a new class of population-based metaheuristic optimization algorithms which represents design variables as a set of qubits and searches a continuous, multi-dimensional landscape through rotation of the qubit's amplitude pair. Every generation rotates those amplitudes toward a single elite, which corresponds to that generation's best. The per-generation cost scales as $O(N_p N_g)$ for $N_p$ chromosomes and $N_g$ genes (decision variables). Production use of such solvers is rarely confined to a single machine class. Prototypes are run on laboratory servers- before moving to rented cloud workstations for more involved campaigns. The largest problems are reserved for leadership-class accelerators. This paper asks whether a single OpenMP~5 source of QIEO, offloaded with \#pragma omp target, is a viable production path in each of those settings. We report three independent, campaigns of the 0/1 knapsack problem against a same-source multi-core Intel CPU baseline. The study comprises approximately 3,000 runs spanning varying chromosome and gene counts, evaluated using both chromosome-level and gene-level offload strategies on the NVIDIA Tesla V100 SXM2, NVIDIA A100 80GB, and AMD Instinct MI300X GPUs. Deployment-specific nuances such as Volta's constant-memory cliffs, Ampere's L2 persistence and cp.async, CDNA~3's Infinity Cache and XCD occupancy are addressed to ensure high performance of these platforms. Results reveal gene-parallel offload achieved geometric-mean speedups of 90$\times$, 136$\times$, and 155$\times$ over a single CPU core on the V100, A100, and MI300X, respectively, and 12$\times$, 17$\times$, and 16.6$\times$ over 72 host threads. Furthermore DetermineElite, the $O(N_p)$ selection of the generation-best chromosome, is found to be better suited to the host than to the device.

We prove the soliton resolution conjecture for the Tzitzéica equation, which is an important integrable model arising in affine differential geometry and characterizing proper affine spheres. Recently, Huang, Wang and Zhu (Math. Ann. 395, 18 (2026)) reported the long-time asymptotic result for the Tzitzéica equation with purely continuous spectrum. Inspired by their work, we carry out a full asymptotic analysis of the Tzitzéica equation. Compared with the previous work, by developing the $\bar{\partial}$-nonlinear steepest descent method, we reveal that the solutions of the Tzitzéica equation exhibits three different asymptotic regions depending on $\xi:=x/t$. For the region $\xi\in(-\infty,-1]\cup[1,\infty)$, where there are no stationary phase points, we rigorously prove that the solution satisfies $u(x,t)=O_\xi(t^{-1})$ for $\xi\ge1$ and $u(x,t)=2\pi i+O_\xi(t^{-1})$ for $\xi\le-1$. In the region $\xi\in(-1,1)$, the phase function $\theta(z)$ has two stationary phase points. The corresponding asymptotic approximations can be characterized with an $N$-soliton solution as well as an interaction term between soliton solutions and the dispersion term with a residual error of order $O(t^{-3/4+b_K})$ with $0\le b_K<1/4$. In the transition regions $\{(x,t):1-\varepsilon<|\xi|<1\}$, the corresponding asymptotic approximation can be characterized by the dispersion term with a residual error of order $O(t^{-3/4+b_\varepsilon})$ where $0\le b_\varepsilon<1/4$. Our results provide a rigorous analysis of the long-time asymptotic behavior of solutions of the Tzitzéica equation in different regions and confirm the soliton resolution conjecture for the Tzitzéica equation.

Lee and Sidford [LS19] showed that the linear program $\min\{c^\top x\ :\ A^\top x=b,\ l\leq x\leq u\}$ over $x\in\mathbb{R}^m$, where $A\in\mathbb{R}^{m\times n}$, can be solved to accuracy $\epsilon$ using $O(\sqrt n\log^{13}m\cdot\log(mU/\epsilon))$ solves of linear systems in $A^\top\mathbf DA$ for positive diagonal matrices $\mathbf D$, where $U$ bounds the magnitudes of the input and of the initial point. Theirs is the first such bound governed by $\mathrm{rank}(A)=n$ rather than by the number of constraints $m$. Song [Son19] mentioned that reducing the $\log^{13}m$ factor is an interesting future direction. Given an interior point and a finite certified magnitude bound $U$, we give an algorithm that uses $O(\sqrt n\log^{4}m\cdot\log(mU/\epsilon))$ solves of such linear systems, improving the Lee–Sidford bound by a factor of $\log^{9}m$. We conjecture that $O(\sqrt n\log(mU/\epsilon))$ such linear-system solves suffice.

Let $(A,I)$ be a henselian pair such that $A$ is a domain of characteristic zero and $I\ne0$. We prove that any flat affine henselian scheme of finite presentation over $(A,I)$ has nonzero first structure-sheaf cohomology whenever one of its fibers over a point in $V(I)$ has positive dimension. This in particular gives a negative answer to the question by Devadas in the mixed-characteristic case. As an application, we also prove that the first structure-sheaf cohomology does not vanish for henselian affinoids of positive dimensions.

We consider some new velocity estimates for general steady Navier-Stokes solutions in plane domains. Namely, we obtain some optimal estimates of the difference between mean values of the velocity over two concentric circles in terms of the $L^2$-norm of the vorticity in the annulus between the circles. The sharpness of the estimates is demonstrated by some unexpected counterexamples.

Coded diffraction patterns (CDPs) provide a structured and physically relevant model for phase retrieval, but the dependence among Fourier measurements generated by a common mask makes sharp stability analysis challenging. For a fixed unit-norm signal $\boldsymbol{x}_\star \in \mathbb{C}^n$, let $\boldsymbol{X}_\star=\boldsymbol{x}_\star\boldsymbol{x}_\star^*$, and let $\mathcal A$ be the lifted CDP measurement operator. We prove that, with $L=O(\log n)$ random masks, the following uniform lower isometry holds with high probability: $\|\boldsymbol{X}-\boldsymbol{X}_\star\|_F \lesssim \log^2(2n) \frac{\|\mathcal A(\boldsymbol{X}-\boldsymbol{X}_\star)\|_2}{\sqrt{nL}}, \boldsymbol{X}\succeq\boldsymbol{0},$ from which we derive two consequences. First, for $\boldsymbol{y}=\mathcal A(\boldsymbol{X}_\star)+\boldsymbol{e}$, PhaseLift-type convex programs achieve the Gaussian-type stable recovery bound $\|\widehat{\boldsymbol{X}}-\boldsymbol{X}_\star\|_F \lesssim \frac{\log^2(2n)}{\sqrt{nL}}\|\boldsymbol{e}\|_2.$ Second, in the noiseless case, the nonconvex factorized loss has a benign landscape when the factor width satisfies $r = O(\log^5(2n))$: every second-order critical point $\boldsymbol{V}\in\mathbb C^{n\times r}$ satisfies $\boldsymbol{V}\boldsymbol{V}^*=\boldsymbol{X}_\star$. The key ingredient is a uniform operator-norm bound over row subsets of the dependent CDP measurement matrix, which permits the removal of a controlled set of adaptively selected rows while preserving tangent injectivity.

We study multiple zeta values in positive characteristic. We construct, using $\infty$-adic multiple zeta values, a coaction of the $\infty$-adic multiple zeta value algebra modulo $\zeta_A(q-1)$ on the $\infty$-adic multiple zeta value algebra itself, and prove that it agrees with Mishiba's coaction constructed via special values of Carlitz multiple polylogarithms. We also determine the coaction on the $\star$-inverse values and show that the antipode on the $\infty$-adic multiple zeta value algebra modulo $\zeta_A(q-1)$ is given by the $\star$-inverse operation. In particular, these results prove Mishiba's conjecture concerning the coaction and the antipode for $\infty$-adic multiple zeta values.

This work addresses the computation of extreme values of observables along trajectories of dynamical systems over an \ks{infinite time horizon}. The setting is intrinsically global: trajectories may exhibit large transient excursions, approach their largest values only asymptotically as time tends to infinity, or escape to infinity in finite time. The proposed framework treats these \mk{ scenarios} in a unified way, so that the resulting extreme values reflect both the transient and the asymptotic behavior of the dynamics. The main difficulty is that neither the time interval nor the state space is assumed to be compact. We address this by combining a radial compactification of the state space with a compactified clock variable and discounted occupation measures. This yields a description of trajectory ensembles on compact sets via occupation measures and the associated Liouville's equation. Extreme values are then characterized through a family of reachability problems for level sets of the observable, which can be solved by a bisection procedure. On the theoretical side, we prove a superposition principle for an infinite-horizon Liouville's equation with a time-distributed terminal measure. This result shows that the measure formulation represents ensembles of stopped admissible trajectories, including trajectories \ks{that never stop}, and establishes equivalence between the trajectory-level and measure-level descriptions.

In many safety-critical applications, control of uncertain dynamical systems relies on observers that estimate states and external disturbances. Neural network observers can improve estimation accuracy, but certifying their Lyapunov stability via Linear Matrix Inequality (LMI) constraints leads to large-scale semidefinite programs (SDPs) that are difficult to solve for large networks. To overcome this scalability bottleneck, we propose a novel two-stage training framework for provably stable neural network observers. Our approach decouples the optimization into a point-guided Lyapunov pre-training phase, which rapidly achieves high estimation accuracy and local stability over sampled states, followed by an LMI fine-tuning phase that efficiently satisfies a strict global Lyapunov stability certificate. We provide formal theoretical guarantees for local stability radii and probabilistic coverage over a prescribed compact error-state domain under specified regularity and sampling assumptions. Experiments on nonlinear control benchmarks and X-29 aircraft ablations show that our LMI-certified neural network observers train significantly faster than direct LMI-based methods and generalize robustly across diverse systems, achieving improved tracking accuracy over a range of observer baselines. The code is available at https://github.com/Berry-Myon/LearningNeuralNetworkObserver.

Every finite nonempty incomplete uniquely decipherable code with maximum word length $k$ has an uncompletable word of length at most $4k^2-3k$. The bound is independent of the number of codewords and their total length. Deleting a complete codeword cycle gives a finite path-counting identity; Kraft equality then supplies a short word of deficient compressed mass. Cyclic averaging and padding turn it into an uncompletable word. Conditional expectation makes the construction polynomial-time and also decides completeness. First-return words extend the bound to mortal families of nonnegative integer $n\times n$ matrices with joint spectral radius at most one, provided every strongly connected component has a vertex meeting every cycle. Such a family has a zero product of length at most $4n^2-3n$. A binary partial deterministic family with $2k-1$ states has shortest zero product of length $k^2+k-1$, establishing the optimal quadratic order. The bounds and the explicit-code algorithm, including its polynomial work bound, are proved in Lean.

Modern service platforms often provide customers with real-time congestion information, such as anticipated waiting times, before they decide whether to use a service. This creates an intertemporal tradeoff in service recommendation: directing a customer to a service may generate immediate value, but the resulting congestion can make that service less attractive to future customers. We study this tradeoff through a finite-horizon stochastic optimization problem in which a platform dynamically recommends among multiple services, each represented by a queue, with customers' joining probabilities decreasing with congestion. We first formulate the problem as a Markov decision process and show that optimal decisions generally depend on the joint congestion state, giving rise to a prohibitively large state space. We develop two complementary approximation algorithms that overcome this curse of dimensionality. First, we provide a quasi-polynomial-time approximation scheme that uses truncation and rounding of the model primitives to compress the joint congestion state, while carefully accounting for how these approximations affect the stochastic evolution of the system to establish near-optimality. Second, we develop a polynomial-time LP-guided algorithm that replaces the joint-state representation with service-level marginal information and achieves a (1-1/e) approximation guarantee under substantially more general service and joining dynamics. Finally, we show that monotone joining behavior, whereby customers become less likely to join as a service becomes more congested, marks a fundamental tractability boundary: without monotonicity, the problem is NP-hard to approximate within any constant factor. This establishes monotonicity as a key behavioral property that enables tractable dynamic service recommendation despite the complexity introduced by congestion-dependent joining.

We investigate the use of neural network solvers for infinity and $p$-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenges associated with large $p$ values, ranging from $2$ to $1000$, on various 2D and 3D domains. Our method offers advantages over traditional physics-based solvers, especially in three dimensions where mesh-based solvers become very costly for these problems. We also establish conditional convergence results for PINN approximations of both problems and a universal approximation result for DeepONet on the parametric $p$-Poisson problem. We demonstrate the effectiveness of these neural network solvers through numerical experiments and compare their performance with conventional methods.

We determine the optimal sample exponent for direct discrete Gaussian SVP search on Haar random unimodular lattices, counting zero outputs. The output is one sampled vector, optionally divided by the greatest common divisor of its lattice coordinates. For every fixed approximation factor $1\leq\gamma<\sqrt e$, the exponent is $\gamma^2/(2e)-\log\gamma$ in natural logarithmic units; it is zero for $\gamma\geq\sqrt e$. For exact SVP this gives $0.2653689\ldots$ in base two. The converse permits arbitrary positive widths chosen from the lattice and all previous outputs, and a fixed width attains every exponent above the threshold. Aggarwal, Dadush, Regev, and Stephens-Davidowitz already give a width within a factor two of optimal for exact SVP on each lattice. We determine the explicit Haar typical rate and show that primitive reduction preserves it. An explicit finite dimensional converse controls all widths simultaneously, including recovery from long multiples; together with finite attainment bounds, it yields query guarantees in both directions. The proof uses random lattice moments and a pointwise Gaussian bound optimized over the width. We account for sampling error and separate sample requirements from generation costs in comparisons with the random lattice search of Pouly and Shen and later algorithms.

We study an inverse problem for a nonlinear Schrödinger equation with a harmonic potential in some spatial directions and free propagation in the remaining directions. The nonlinearity has an unknown power and a real, spatially dependent coefficient. For small incoming data in a weighted energy space, we construct the wave operator and show that it determines both the power and the coefficient when the latter depends either on the confined variables or on the free variables. The proof uses the first nonlinear term of the wave operator and concentrated product data. We explain why dependence on the free variables requires a stronger lower bound on the power. We also discuss the outgoing wave operator, the small-data scattering operator, and coefficients depending on all spatial variables.

We give explicit algorithms that factor elements of the standard diagram monoids into their usual generators. These algorithms generalize sorting from permutations to partial matchings and set partitions. In every case the worst-case complexity is $n^2$, which is optimal for algorithms that explicitly list the factors. We also determine the average complexity of our algorithms.

We prove that closed Einstein four-manifolds $(M,g)$ with positive sectional curvature and Euler characteristic and signature satisfying $2\chi(M)-3|\tau(M)|\le4$ must be homothetically isometric to the round $S^4$ or to $\mathbb{CP}^2$ with the Fubini-Study metric. The proof relies on an upper bound on the ADM mass of conformal blow-ups of $(M,g)$, combined with a lower one from \cite{GM}.

In this article, we first give a characterization of hyponormal Toeplitz operators with operator valued normal symbols. Latter, we examine when the class of hyponormal Toeplitz operators with operator valued symbols are analytic or normal.

We consider the recovery of the fractional order in a time-fractional evolution equation with a source of the form $p(t)f$, where the temporal factor $p$ is also unknown but belongs to a bounded admissible class and has a prescribed nonzero value at $t=0$. From one scalar observation we prove a uniform small-time expansion and obtain a Hölder estimate for the fractional order. More precisely, if a fixed derivative order $\gamma$ lies below every admissible fractional order, then the stability exponent is $\alpha_+/(2\alpha_+-\gamma)$; for the undifferentiated observation this gives the exponent $1/2$. We also show that differentiating the data to an order larger than all admissible fractional orders is supercritical: the corresponding derivative is unbounded near $t=0$ whenever the two fractional orders are different. A uniform estimate for the two-parameter Mittag–Leffler function, including the diagonal case, is proved in the appendix.

We consider inverse problems for a time-fractional diffusion equation with a second-order elliptic operator and the homogeneous Dirichlet boundary condition in a bounded domain of $\mathbb R^N$. The initial value is unknown and is not assumed to be zero. The main result concerns the simultaneous determination of the fractional order and the time-dependent source factor from one weighted spatial observation. For irrational orders in $(0,1)$ and bounded source factors satisfying an exponential integrability condition, we prove that a nonzero observation given for all positive times uniquely determines both unknowns. We also describe the information about the initial value determined by this observation. A counterexample shows that uniqueness of the source factor may fail at the classical order $\alpha=1$, even when the observation is nonzero. In the second part, we establish two uniqueness results for the fractional order when the elliptic operators, initial values, and source terms in the compared equations may be different. The first result is proved by the Laplace transform, whereas the second uses short-time observations and the asymptotic behavior of the Mittag–Leffler functions.

We consider the Korteweg-de Vries (KdV) approximation for the good Boussinesq equation. Building upon the low-regularity local-in-time justification established by Hong and Yang \cite{HY2024}, we prove that this approximation remains valid on logarithmically long time intervals. The key ingredient is a persistence of regularity argument, which allows the local approximation to be iterated using the rescaled conservation law for the Boussinesq equation.

Let $\mathcal{R}$ be the hyperfinite $\mathrm{II}_1$ factor. Considering the partial sum operators $(\mathcal{S}_n)_{n\geq 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c>0$ such that \begin{equation*} \sup_{n\geq1}\|\mathcal{S}_n(f)\|_{L_{1,\infty}(\mathcal{R})} \leq c\|f\|_{L_1(\mathcal{R})},\quad f \in L_1(\mathcal{R}), \end{equation*} and, for every $1<p<\infty$, $$\sup_{n\geq1}\|\mathcal{S}_n(f)\|_{L_p(\mathcal{R})} \leq c\frac{p}{p-1}\|f\|_{L_p(\mathcal{R})},\quad f \in L_p(\mathcal{R}).$$ Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calderón-Zygmund decomposition established in \cite{CCP2022}. Consequently, we resolve the problem of weak type $(1,1)$ estimate communicated to the authors by Fedor Sukochev, and substantially improve the strong type (p,p) estimates obtained in \cite{DFdePS2001} by achieving the optimal order $\frac{p}{p-1}$.

For $a$ in $\mathbb{R}$, a subset $V$ contained in $\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \in V \Longrightarrow a x + (1-a) y \in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\mathbb{R}^{n}$ that contain $V$, and Pinch also defines $D(a)$ as the $a$-convex hull of $\{ 0, 1 \}$ in $\mathbb{R}^{1}$. Pinch conjectured that if $a$ is a totally real algebraic integer and $D(a)$ has no limit points, then every algebraic conjugate of $a$ other than $a$ is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.

We consider a inverse problem for a hyperbolic equation. The objective is to simultaneous recover both the spatially varying coefficient, whose square root represents the wave speed, and the initial displacement from a single pair of Dirichlet-Neumann Cauchy data. By introducing a suitable transformation and differentiating the resulting equation in time, we derive a coupled system for the differences of the unknown coefficient and initial displacement. Combining weighted energy estimates with Carleman estimates, we establish a Lipschitz stability estimate for both unknown quantities in appropriate Sobolev norms. The proof relies on a priori regularity assumptions, non-degeneracy conditions on the initial data, and structural inequalities controlling the interaction between the coefficient and initial-displacement differences. Our result extends coefficient-only stability results to a simultaneous coefficient–initial-displacement identification problem.

In this paper, we investigate transition fronts and spreading solutions of Fisher–KPP equations in one-dimensional almost periodic media, both on the real line and on the lattice. Let $\lambda_1$ be the supremum of the spectrum of the linearized operator acting on $L^2(\R)$ or $\ell^2(\Z)$, respectively, and let \(L(\lambda_1)\) denote the spatial Lyapunov exponent at the spectral parameter \(\lambda_1\). We prove that, if $L(\lambda_1)>0$, the global mean speed of any transition front is at most $\lambda_1/L(\lambda_1)$. Moreover, if a solution of the Cauchy problem spreads faster than this bound, its transition width is unbounded along a sequence of times. This occurs for a class of initial data with slowly decaying exponential tails. This contrasts with periodic media, where pulsating fronts exist at every speed above the minimal speed. As applications, we consider almost Mathieu coefficients and a continuous quasiperiodic coefficient with two frequencies. Together with the results of Nadin–Rossi and Liang–Wang–Zhou–Zhou \citep{LWZZ24}, our results provide an almost complete picture of the admissible speeds of generalized transition fronts, while leaving the critical cases open.

We propose a Sparsely Connected Rank-Inspired Neural Network (SCRINN) for function approximation and the numerical solution of partial differential equations (PDEs). By introducing a structured sparse connectivity architecture and an orthogonality- and normalization-guided loss function, SCRINN constructs an approximately orthonormal neural basis while promoting progressive rank growth of its evaluation matrix and reducing the number of trainable parameters. A two-stage framework is developed, in which an approximately orthonormal neural basis is first constructed, and the output weights are subsequently determined using least-squares methods. Numerical experiments on function approximation and a variety of steady-state, time-dependent, and nonlinear PDEs demonstrate that SCRINN consistently achieves high approximation accuracy with substantially lower training costs. The code is publicly available on GitHub.