Real-time estimation of full unsteady aerodynamic flow fields $\mathbf{u}(\mathbf{x}, t) = (u, v)$ from ultra-sparse discrete sensor observations (e.g., $M \le 10$ point velocity probes across thousands of grid cells) is a foundational challenge in active flow control, wind tunnel telemetry, and unmanned aerial vehicle (UAV) gust alleviation. Physics-Informed Neural Networks (PINNs) have emerged as a dominant scientific machine learning paradigm; however, for online cold-start inverse reconstruction—where an unexpected, non-stationary flow snapshot must be inverted on-the-fly without prior model fine-tuning—PINNs require thousands of Adam/L-BFGS backpropagation iterations ($8.33\text{ s}$ per state), violating 1~kHz real-time deadlines by four orders of magnitude. While offline pre-trained neural surrogates evaluate rapidly, they demand extensive parameter sweeps and inherently suffer from non-physical divergence leakage and boundary penetration due to soft penalty loss formulations. Similarly, classical Gappy-POD provides an instantaneous linear estimate ($<0.05\text{ ms}$), yet modal truncation invariably leaks divergence ($\|\nabla \cdot \mathbf{u}\|_\infty \approx 1.84 \times 10^{-2}\text{ s}^{-1}$) and violates wall adherence. In this paper, we propose a two-tier discrete-continuous architecture unifying Deep Permutation Networks (DPN) with algebraic idempotent metric projections ($\boldsymbol{\Pi}^2 = \boldsymbol{\Pi}$) and sequential Projection Onto Convex Sets (POCS). In Tier 1, a discrete APM limit-cycle oscillator tracks the non-linear vortex shedding phase and filters modal coefficients along the invariant limit-cycle attractor in $\mathcal{O}(1)$ time with zero backpropagation. In Tier 2, four idempotent projectors cyclically interleave: An orthonormal proper orthogonal decomposition (POD) subspace projector $\mathbf{P}_{\text{POD}} = \boldsymbol{\Phi}\boldsymbol{\Phi}^\dagger$; A discrete Hodge-Helmholtz divergence-free projector $\boldsymbol{\Pi}_{\text{div}}$ enforcing $\nabla \cdot \mathbf{u} = 0$; A no-slip wall projector $\boldsymbol{\Pi}_{\text{wall}}$; and A Dirichlet sensor clamping projector $\boldsymbol{\Pi}_{\text{sensor}}$. In rigorous head-to-head empirical testing against PyTorch PINN and Gappy-POD on a Navier-Stokes Direct Numerical Simulation (DNS) wake ($Re=100$, $St=0.160$, 2048 cells, 10 sensors representing $0.49\%$ spatial coverage), the proposed DPN-IdemSolver achieves: An online execution latency of $0.275\text{ ms}$ ($275\ \mu\text{s}$), representing a $30,295\times$ speedup over online PINN backpropagation ($8,329.6\text{ ms}$); A reconstruction accuracy of $R^2 = 99.99\%$ (compared to $96.22\%$ for Gappy-POD and $69.88\%$ for PINN); A relative velocity $L_2$ error of $0.44\%$ ($11\times$ lower than Gappy-POD and $47.5\times$ lower than PINN); Machine-precision solenoidal compliance ($\|\nabla \cdot \mathbf{u}\|_\infty = 1.26 \times 10^{-11}\text{ s}^{-1}$); $3\times$ noise suppression under realistic probe turbulence; and Strictly zero GPU and zero neural network backpropagation training, operating within $5\text{ MB}$ of RAM on standard edge computing nodes.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
For four decades, deep artificial neural networks have relied on continuous vector spaces, dense floating-point General Matrix Multiplications (GEMM: hl+1 = σ(Wlhl + bl)), and gradient-based backpropagation. While effective in data centers, this paradigm incurs severe thermodynamic, memory bandwidth, and numerical stability bottlenecks in ultra-low-power edge computing, implantable micro-sensors, and real-time robotic controllers. We formulate Deep Permutation Networks (DPN), an entirely discrete, topological learning paradigm operating over stratified symmetric groups and factorized Young subgroups Sλ = Sλ1 × ... × SλK. Originating from foundational theoretical discoveries on invariant sub-manifolds, DPN completely eliminates continuous GEMMs and backpropagation. Continuous multi-modal data streams are ingested via a Unified Parallel-Branch Ingestion Architecture (ParallelBranchDPNClassifier): a core Young subgroup ordinal sorting branch captures monotonic order invariants, while parallel spectral, variational, and geometric filter branches—incorporating 3D/2D/1D discrete Laplace-Beltrami operators, Fast Walsh-Hadamard transforms, Takens delay-permutation embeddings, Chebyshev orthogonal projections, AC Fourier harmonics, and selectable foreground bounding-box isolation—evaluate feature representations with strictly 0.00 arithmetic MACs. Each neural layer acts as an invariant algebraic permutation manifold propagating activations as zero-FLOP phase rotations, while inter-layer communication is governed by cycle splicing in O(1) time. Training executes without chain-rule gradients via single-pass topological transposition steering into target Tarski attractor basins (Π2 = Π). When evaluated via post-synthesis standard-cell gate-level simulations in TSMC 28 nm HPC+ CMOS (modeled via CACTI 7.0 and Cadence Genus), a 3-stage pipelined APM-RAM controller operates at 4.5 GHz with 0.52–1.12 ns latency, dissipating only 6.6 μW – 23.4 μW and requiring 14.8–48.2 fJ per inference (>15,000× – 40,000× lower energy than iso-process systolic GEMM accelerators). Extensive empirical evaluations across diverse physical and digital modalities—including 3D LiDAR point clouds, 3D volumetric medical CT, industrial 2D laser triangulation profilers, acoustic speech resonances, 3D RGB-D spatial scenes, 16-channel BCI EEG telemetry, UCI benchmarks, and dense text embeddings—confirm strictly 0.00 arithmetic MACs, high classification accuracy, sub-kilobyte static memory footprints (1.8–24.6 KB), and exact mathematical determinism.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
For four decades, deep artificial neural networks have relied on continuous vector spaces, dense floating-point General Matrix Multiplications (GEMM: hl+1 = σ(Wlhl + bl)), and gradient-based backpropagation. While effective in data centers, this paradigm incurs severe thermodynamic, memory bandwidth, and numerical stability bottlenecks in ultra-low-power edge computing, implantable micro-sensors, and real-time robotic controllers. We formulate Deep Permutation Networks (DPN), an entirely discrete, topological learning paradigm operating over stratified symmetric groups and factorized Young subgroups Sλ = Sλ1 × ... × SλK. Originating from foundational theoretical discoveries on invariant sub-manifolds, DPN completely eliminates continuous GEMMs and backpropagation. Continuous multi-modal data streams are ingested via a Unified Parallel-Branch Ingestion Architecture (ParallelBranchDPNClassifier): a core Young subgroup ordinal sorting branch captures monotonic order invariants, while parallel spectral, variational, and geometric filter branches—incorporating 3D/2D/1D discrete Laplace-Beltrami operators, Fast Walsh-Hadamard transforms, Takens delay-permutation embeddings, Chebyshev orthogonal projections, AC Fourier harmonics, and selectable foreground bounding-box isolation—evaluate feature representations with strictly 0.00 arithmetic MACs. Each neural layer acts as an invariant algebraic permutation manifold propagating activations as zero-FLOP phase rotations, while inter-layer communication is governed by cycle splicing in O(1) time. Training executes without chain-rule gradients via single-pass topological transposition steering into target Tarski attractor basins (Π2 = Π). When evaluated via post-synthesis standard-cell gate-level simulations in TSMC 28 nm HPC+ CMOS (modeled via CACTI 7.0 and Cadence Genus), a 3-stage pipelined APM-RAM controller operates at 4.5 GHz with 0.52–1.12 ns latency, dissipating only 6.6 μW – 23.4 μW and requiring 14.8–48.2 fJ per inference (>15,000× – 40,000× lower energy than iso-process systolic GEMM accelerators). Extensive empirical evaluations across diverse physical and digital modalities—including 3D LiDAR point clouds, 3D volumetric medical CT, industrial 2D laser triangulation profilers, acoustic speech resonances, 3D RGB-D spatial scenes, 16-channel BCI EEG telemetry, UCI benchmarks, and dense text embeddings—confirm strictly 0.00 arithmetic MACs, high classification accuracy, sub-kilobyte static memory footprints (1.8–24.6 KB), and exact mathematical determinism.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
Precise on-board estimation of State of Charge (SoC) and State of Health (SoH) in lithium-ion battery management systems (BMS) remains a critical bottleneck in electric vehicle range predictability, fast-charging safety, and pack longevity. Conventional BMS firmware is polarized between two inadequate paradigms: empirical Equivalent Circuit Models (ECM) paired with Extended Kalman Filters (EKF), which evaluate rapidly but suffer from cumulative sensor bias drift (2%–6% SoC error over typical drive cycles) and lack physical diffusion dynamics during pulse acceleration; and full Doyle-Fuller-Newman (DFN/P2D) electrochemical models, whose coupled nonlinear differential-algebraic equations (DAEs) require iterative Newton-Raphson solvers and multi-second compute budgets that exceed the memory and compute capacity of automotive microcontrollers (e.g., ARM Cortex-M4/M7, Infineon AURIX TC397). In this paper, we resolve this dichotomy by introducing IdemBattery, a deterministic, zero-heap electrochemical estimator based on an Enhanced Single Particle Model with Thermal and Degradation Coupling (SPM-T) accelerated by algebraic idempotent metric projectors (Π2 = Π). Solid-phase lithium transport is discretized via conservative spherical finite volumes and inverted in closed form using an unconditionally stable O(N) tridiagonal Thomas algorithm. Interfacial charge transfer is evaluated analytically via inverse hyperbolic sine Butler-Volmer kinetics with second-order boundary layer flux extrapolation, completely bypassing nonlinear root-finding. To eliminate numerical discretization truncation leakage, we formulate the Volume-Weighted Lithium Mass Projector (ΠmassL2), which rigorously satisfies exact mass conservation while provably preserving internal spatial concentration gradients: ∂/∂r(ΠmassL2 cs) ≡ ∂cs/∂r. Boundary lattice saturation is resolved simultaneously via an exact O(N) continuous quadratic knapsack projector (Mmass ∩ Mbox). Physical current-sensor zero-offset bias (+80 mA) is rejected via a closed-loop observer (IdemObserver) with terminal voltage innovation feedback. Parametrized with peer-reviewed experimental data from Chen et al. (2020) for NMC-811/Graphite (LG M50) and Safari & Delacourt (2011) for LFP/Graphite, and incorporating Pinson-Bazant SEI degradation, IdemBattery achieves: An average execution latency of 60.49 μs per step on host CPU (σ = 48.34 μs, P99 = 148.91 μs, ≈ 15,000 instruction cycles on automotive MCUs, comfortably fitting within 10–100 ms BMS loop deadlines); Provable machine-precision lithium mass invariance (|ΔQ| / Q0 < 10−14); Total dynamic rejection of sensor bias drift; and Strictly zero dynamic heap memory allocation (0.0 bytes), making it architecturally compatible with safety-critical automotive Electronic Control Units (ECUs).
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
For four decades, deep artificial neural networks have relied on continuous vector spaces, dense floating-point General Matrix Multiplications (GEMM: hl+1 = σ(Wlhl + bl)), and gradient-based backpropagation. While effective in data centers, this paradigm incurs severe thermodynamic, memory bandwidth, and numerical stability bottlenecks in ultra-low-power edge computing, implantable micro-sensors, and real-time robotic controllers. We formulate Deep Permutation Networks (DPN), an entirely discrete, topological learning paradigm operating over stratified symmetric groups and factorized Young subgroups Sλ = Sλ1 × ... × SλK. Originating from foundational theoretical discoveries on invariant sub-manifolds, DPN completely eliminates continuous GEMMs and backpropagation. Continuous multi-modal data streams are ingested via a Unified Parallel-Branch Ingestion Architecture (ParallelBranchDPNClassifier): a core Young subgroup ordinal sorting branch captures monotonic order invariants, while parallel spectral, variational, and geometric filter branches—incorporating 3D/2D/1D discrete Laplace-Beltrami operators, Fast Walsh-Hadamard transforms, Takens delay-permutation embeddings, Chebyshev orthogonal projections, AC Fourier harmonics, and selectable foreground bounding-box isolation—evaluate feature representations with strictly 0.00 arithmetic MACs. Each neural layer acts as an invariant algebraic permutation manifold propagating activations as zero-FLOP phase rotations, while inter-layer communication is governed by cycle splicing in O(1) time. Training executes without chain-rule gradients via single-pass topological transposition steering into target Tarski attractor basins (Π2 = Π). When evaluated via post-synthesis standard-cell gate-level simulations in TSMC 28 nm HPC+ CMOS (modeled via CACTI 7.0 and Cadence Genus), a 3-stage pipelined APM-RAM controller operates at 4.5 GHz with 0.52–1.12 ns latency, dissipating only 6.6 μW – 23.4 μW and requiring 14.8–48.2 fJ per inference (>15,000× – 40,000× lower energy than iso-process systolic GEMM accelerators). Extensive empirical evaluations across diverse physical and digital modalities—including 3D LiDAR point clouds, 3D volumetric medical CT, industrial 2D laser triangulation profilers, acoustic speech resonances, 3D RGB-D spatial scenes, 16-channel BCI EEG telemetry, UCI benchmarks, and dense text embeddings—confirm strictly 0.00 arithmetic MACs, high classification accuracy, sub-kilobyte static memory footprints (1.8–24.6 KB), and exact mathematical determinism.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
Real-time estimation of full unsteady aerodynamic flow fields $\mathbf{u}(\mathbf{x}, t) = (u, v)$ from ultra-sparse discrete sensor observations (e.g., $M \le 10$ point velocity probes across thousands of grid cells) is a foundational challenge in active flow control, wind tunnel telemetry, and unmanned aerial vehicle (UAV) gust alleviation. Physics-Informed Neural Networks (PINNs) have emerged as a dominant scientific machine learning paradigm; however, for online cold-start inverse reconstruction—where an unexpected, non-stationary flow snapshot must be inverted on-the-fly without prior model fine-tuning—PINNs require thousands of Adam/L-BFGS backpropagation iterations ($8.33\text{ s}$ per state), violating 1~kHz real-time deadlines by four orders of magnitude. While offline pre-trained neural surrogates evaluate rapidly, they demand extensive parameter sweeps and inherently suffer from non-physical divergence leakage and boundary penetration due to soft penalty loss formulations. Similarly, classical Gappy-POD provides an instantaneous linear estimate ($<0.05\text{ ms}$), yet modal truncation invariably leaks divergence ($\|\nabla \cdot \mathbf{u}\|_\infty \approx 1.84 \times 10^{-2}\text{ s}^{-1}$) and violates wall adherence. In this paper, we propose a two-tier discrete-continuous architecture unifying Deep Permutation Networks (DPN) with algebraic idempotent metric projections ($\boldsymbol{\Pi}^2 = \boldsymbol{\Pi}$) and sequential Projection Onto Convex Sets (POCS). In Tier 1, a discrete APM limit-cycle oscillator tracks the non-linear vortex shedding phase and filters modal coefficients along the invariant limit-cycle attractor in $\mathcal{O}(1)$ time with zero backpropagation. In Tier 2, four idempotent projectors cyclically interleave: An orthonormal proper orthogonal decomposition (POD) subspace projector $\mathbf{P}_{\text{POD}} = \boldsymbol{\Phi}\boldsymbol{\Phi}^\dagger$; A discrete Hodge-Helmholtz divergence-free projector $\boldsymbol{\Pi}_{\text{div}}$ enforcing $\nabla \cdot \mathbf{u} = 0$; A no-slip wall projector $\boldsymbol{\Pi}_{\text{wall}}$; and A Dirichlet sensor clamping projector $\boldsymbol{\Pi}_{\text{sensor}}$. In rigorous head-to-head empirical testing against PyTorch PINN and Gappy-POD on a Navier-Stokes Direct Numerical Simulation (DNS) wake ($Re=100$, $St=0.160$, 2048 cells, 10 sensors representing $0.49\%$ spatial coverage), the proposed DPN-IdemSolver achieves: An online execution latency of $0.275\text{ ms}$ ($275\ \mu\text{s}$), representing a $30,295\times$ speedup over online PINN backpropagation ($8,329.6\text{ ms}$); A reconstruction accuracy of $R^2 = 99.99\%$ (compared to $96.22\%$ for Gappy-POD and $69.88\%$ for PINN); A relative velocity $L_2$ error of $0.44\%$ ($11\times$ lower than Gappy-POD and $47.5\times$ lower than PINN); Machine-precision solenoidal compliance ($\|\nabla \cdot \mathbf{u}\|_\infty = 1.26 \times 10^{-11}\text{ s}^{-1}$); $3\times$ noise suppression under realistic probe turbulence; and Strictly zero GPU and zero neural network backpropagation training, operating within $5\text{ MB}$ of RAM on standard edge computing nodes.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
Precise on-board estimation of State of Charge (SoC) and State of Health (SoH) in lithium-ion battery management systems (BMS) remains a critical bottleneck in electric vehicle range predictability, fast-charging safety, and pack longevity. Conventional BMS firmware is polarized between two inadequate paradigms: empirical Equivalent Circuit Models (ECM) paired with Extended Kalman Filters (EKF), which evaluate rapidly but suffer from cumulative sensor bias drift (2%–6% SoC error over typical drive cycles) and lack physical diffusion dynamics during pulse acceleration; and full Doyle-Fuller-Newman (DFN/P2D) electrochemical models, whose coupled nonlinear differential-algebraic equations (DAEs) require iterative Newton-Raphson solvers and multi-second compute budgets that exceed the memory and compute capacity of automotive microcontrollers (e.g., ARM Cortex-M4/M7, Infineon AURIX TC397). In this paper, we resolve this dichotomy by introducing IdemBattery, a deterministic, zero-heap electrochemical estimator based on an Enhanced Single Particle Model with Thermal and Degradation Coupling (SPM-T) accelerated by algebraic idempotent metric projectors (Π2 = Π). Solid-phase lithium transport is discretized via conservative spherical finite volumes and inverted in closed form using an unconditionally stable O(N) tridiagonal Thomas algorithm. Interfacial charge transfer is evaluated analytically via inverse hyperbolic sine Butler-Volmer kinetics with second-order boundary layer flux extrapolation, completely bypassing nonlinear root-finding. To eliminate numerical discretization truncation leakage, we formulate the Volume-Weighted Lithium Mass Projector (ΠmassL2), which rigorously satisfies exact mass conservation while provably preserving internal spatial concentration gradients: ∂/∂r(ΠmassL2 cs) ≡ ∂cs/∂r. Boundary lattice saturation is resolved simultaneously via an exact O(N) continuous quadratic knapsack projector (Mmass ∩ Mbox). Physical current-sensor zero-offset bias (+80 mA) is rejected via a closed-loop observer (IdemObserver) with terminal voltage innovation feedback. Parametrized with peer-reviewed experimental data from Chen et al. (2020) for NMC-811/Graphite (LG M50) and Safari & Delacourt (2011) for LFP/Graphite, and incorporating Pinson-Bazant SEI degradation, IdemBattery achieves: An average execution latency of 60.49 μs per step on host CPU (σ = 48.34 μs, P99 = 148.91 μs, ≈ 15,000 instruction cycles on automotive MCUs, comfortably fitting within 10–100 ms BMS loop deadlines); Provable machine-precision lithium mass invariance (|ΔQ| / Q0 < 10−14); Total dynamic rejection of sensor bias drift; and Strictly zero dynamic heap memory allocation (0.0 bytes), making it architecturally compatible with safety-critical automotive Electronic Control Units (ECUs).
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
Precise on-board estimation of State of Charge (SoC) and State of Health (SoH) in lithium-ion battery management systems (BMS) remains a critical bottleneck in electric vehicle range predictability, fast-charging safety, and pack longevity. Conventional BMS firmware is polarized between two inadequate paradigms: empirical Equivalent Circuit Models (ECM) paired with Extended Kalman Filters (EKF), which evaluate rapidly but suffer from cumulative sensor bias drift (2%–6% SoC error over typical drive cycles) and lack physical diffusion dynamics during pulse acceleration; and full Doyle-Fuller-Newman (DFN/P2D) electrochemical models, whose coupled nonlinear differential-algebraic equations (DAEs) require iterative Newton-Raphson solvers and multi-second compute budgets that exceed the memory and compute capacity of automotive microcontrollers (e.g., ARM Cortex-M4/M7, Infineon AURIX TC397). In this paper, we resolve this dichotomy by introducing IdemBattery, a deterministic, zero-heap electrochemical estimator based on an Enhanced Single Particle Model with Thermal and Degradation Coupling (SPM-T) accelerated by algebraic idempotent metric projectors (Π2 = Π). Solid-phase lithium transport is discretized via conservative spherical finite volumes and inverted in closed form using an unconditionally stable O(N) tridiagonal Thomas algorithm. Interfacial charge transfer is evaluated analytically via inverse hyperbolic sine Butler-Volmer kinetics with second-order boundary layer flux extrapolation, completely bypassing nonlinear root-finding. To eliminate numerical discretization truncation leakage, we formulate the Volume-Weighted Lithium Mass Projector (ΠmassL2), which rigorously satisfies exact mass conservation while provably preserving internal spatial concentration gradients: ∂/∂r(ΠmassL2 cs) ≡ ∂cs/∂r. Boundary lattice saturation is resolved simultaneously via an exact O(N) continuous quadratic knapsack projector (Mmass ∩ Mbox). Physical current-sensor zero-offset bias (+80 mA) is rejected via a closed-loop observer (IdemObserver) with terminal voltage innovation feedback. Parametrized with peer-reviewed experimental data from Chen et al. (2020) for NMC-811/Graphite (LG M50) and Safari & Delacourt (2011) for LFP/Graphite, and incorporating Pinson-Bazant SEI degradation, IdemBattery achieves: An average execution latency of 60.49 μs per step on host CPU (σ = 48.34 μs, P99 = 148.91 μs, ≈ 15,000 instruction cycles on automotive MCUs, comfortably fitting within 10–100 ms BMS loop deadlines); Provable machine-precision lithium mass invariance (|ΔQ| / Q0 < 10−14); Total dynamic rejection of sensor bias drift; and Strictly zero dynamic heap memory allocation (0.0 bytes), making it architecturally compatible with safety-critical automotive Electronic Control Units (ECUs).
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
Real-time estimation of full unsteady aerodynamic flow fields $\mathbf{u}(\mathbf{x}, t) = (u, v)$ from ultra-sparse discrete sensor observations (e.g., $M \le 10$ point velocity probes across thousands of grid cells) is a foundational challenge in active flow control, wind tunnel telemetry, and unmanned aerial vehicle (UAV) gust alleviation. Physics-Informed Neural Networks (PINNs) have emerged as a dominant scientific machine learning paradigm; however, for online cold-start inverse reconstruction—where an unexpected, non-stationary flow snapshot must be inverted on-the-fly without prior model fine-tuning—PINNs require thousands of Adam/L-BFGS backpropagation iterations ($8.33\text{ s}$ per state), violating 1~kHz real-time deadlines by four orders of magnitude. While offline pre-trained neural surrogates evaluate rapidly, they demand extensive parameter sweeps and inherently suffer from non-physical divergence leakage and boundary penetration due to soft penalty loss formulations. Similarly, classical Gappy-POD provides an instantaneous linear estimate ($<0.05\text{ ms}$), yet modal truncation invariably leaks divergence ($\|\nabla \cdot \mathbf{u}\|_\infty \approx 1.84 \times 10^{-2}\text{ s}^{-1}$) and violates wall adherence. In this paper, we propose a two-tier discrete-continuous architecture unifying Deep Permutation Networks (DPN) with algebraic idempotent metric projections ($\boldsymbol{\Pi}^2 = \boldsymbol{\Pi}$) and sequential Projection Onto Convex Sets (POCS). In Tier 1, a discrete APM limit-cycle oscillator tracks the non-linear vortex shedding phase and filters modal coefficients along the invariant limit-cycle attractor in $\mathcal{O}(1)$ time with zero backpropagation. In Tier 2, four idempotent projectors cyclically interleave: An orthonormal proper orthogonal decomposition (POD) subspace projector $\mathbf{P}_{\text{POD}} = \boldsymbol{\Phi}\boldsymbol{\Phi}^\dagger$; A discrete Hodge-Helmholtz divergence-free projector $\boldsymbol{\Pi}_{\text{div}}$ enforcing $\nabla \cdot \mathbf{u} = 0$; A no-slip wall projector $\boldsymbol{\Pi}_{\text{wall}}$; and A Dirichlet sensor clamping projector $\boldsymbol{\Pi}_{\text{sensor}}$. In rigorous head-to-head empirical testing against PyTorch PINN and Gappy-POD on a Navier-Stokes Direct Numerical Simulation (DNS) wake ($Re=100$, $St=0.160$, 2048 cells, 10 sensors representing $0.49\%$ spatial coverage), the proposed DPN-IdemSolver achieves: An online execution latency of $0.275\text{ ms}$ ($275\ \mu\text{s}$), representing a $30,295\times$ speedup over online PINN backpropagation ($8,329.6\text{ ms}$); A reconstruction accuracy of $R^2 = 99.99\%$ (compared to $96.22\%$ for Gappy-POD and $69.88\%$ for PINN); A relative velocity $L_2$ error of $0.44\%$ ($11\times$ lower than Gappy-POD and $47.5\times$ lower than PINN); Machine-precision solenoidal compliance ($\|\nabla \cdot \mathbf{u}\|_\infty = 1.26 \times 10^{-11}\text{ s}^{-1}$); $3\times$ noise suppression under realistic probe turbulence; and Strictly zero GPU and zero neural network backpropagation training, operating within $5\text{ MB}$ of RAM on standard edge computing nodes.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
Real-time estimation of full unsteady aerodynamic flow fields $\mathbf{u}(\mathbf{x}, t) = (u, v)$ from ultra-sparse discrete sensor observations (e.g., $M \le 10$ point velocity probes across thousands of grid cells) is a foundational challenge in active flow control, wind tunnel telemetry, and unmanned aerial vehicle (UAV) gust alleviation. Physics-Informed Neural Networks (PINNs) have emerged as a dominant scientific machine learning paradigm; however, for online cold-start inverse reconstruction—where an unexpected, non-stationary flow snapshot must be inverted on-the-fly without prior model fine-tuning—PINNs require thousands of Adam/L-BFGS backpropagation iterations ($8.33\text{ s}$ per state), violating 1~kHz real-time deadlines by four orders of magnitude. While offline pre-trained neural surrogates evaluate rapidly, they demand extensive parameter sweeps and inherently suffer from non-physical divergence leakage and boundary penetration due to soft penalty loss formulations. Similarly, classical Gappy-POD provides an instantaneous linear estimate ($<0.05\text{ ms}$), yet modal truncation invariably leaks divergence ($\|\nabla \cdot \mathbf{u}\|_\infty \approx 1.84 \times 10^{-2}\text{ s}^{-1}$) and violates wall adherence. In this paper, we propose a two-tier discrete-continuous architecture unifying Deep Permutation Networks (DPN) with algebraic idempotent metric projections ($\boldsymbol{\Pi}^2 = \boldsymbol{\Pi}$) and sequential Projection Onto Convex Sets (POCS). In Tier 1, a discrete APM limit-cycle oscillator tracks the non-linear vortex shedding phase and filters modal coefficients along the invariant limit-cycle attractor in $\mathcal{O}(1)$ time with zero backpropagation. In Tier 2, four idempotent projectors cyclically interleave: An orthonormal proper orthogonal decomposition (POD) subspace projector $\mathbf{P}_{\text{POD}} = \boldsymbol{\Phi}\boldsymbol{\Phi}^\dagger$; A discrete Hodge-Helmholtz divergence-free projector $\boldsymbol{\Pi}_{\text{div}}$ enforcing $\nabla \cdot \mathbf{u} = 0$; A no-slip wall projector $\boldsymbol{\Pi}_{\text{wall}}$; and A Dirichlet sensor clamping projector $\boldsymbol{\Pi}_{\text{sensor}}$. In rigorous head-to-head empirical testing against PyTorch PINN and Gappy-POD on a Navier-Stokes Direct Numerical Simulation (DNS) wake ($Re=100$, $St=0.160$, 2048 cells, 10 sensors representing $0.49\%$ spatial coverage), the proposed DPN-IdemSolver achieves: An online execution latency of $0.275\text{ ms}$ ($275\ \mu\text{s}$), representing a $30,295\times$ speedup over online PINN backpropagation ($8,329.6\text{ ms}$); A reconstruction accuracy of $R^2 = 99.99\%$ (compared to $96.22\%$ for Gappy-POD and $69.88\%$ for PINN); A relative velocity $L_2$ error of $0.44\%$ ($11\times$ lower than Gappy-POD and $47.5\times$ lower than PINN); Machine-precision solenoidal compliance ($\|\nabla \cdot \mathbf{u}\|_\infty = 1.26 \times 10^{-11}\text{ s}^{-1}$); $3\times$ noise suppression under realistic probe turbulence; and Strictly zero GPU and zero neural network backpropagation training, operating within $5\text{ MB}$ of RAM on standard edge computing nodes.
A. Emre Cetin· Zenodo (CERN European Organi...· 0 citations
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