# Physics-Informed Neural Networks

Starting from noisy velocity images, unsteady three-dimensional blood flow and elastic vessel-wall motion can be reconstructed together without a prescribed inlet waveform or initial velocity. The solver alternates a fluid update, in which a physics-informed network fits the observations while enforcing the incompressible Navier–Stokes equations on the moving domain, with a geometry update, in which registration plus interface traction continuity and an elastic momentum residual constrain wall displacement. On a converging channel and a patient-like aorta, the PirateNets version reaches velocity relative errors of 0.0243 and 0.0372, below the 0.1066 and 0.1305 of a standard PINN of the same depth and width.

Starting from noisy velocity images, unsteady three-dimensional blood flow and elastic vessel-wall motion can be reconstructed together without a prescribed inlet waveform or initial velocity. The solver alternates a fluid update, in which a physics-informed network fits the observations while enforcing the incompressible Navier–Stokes equations on the moving domain, with a geometry update, in which registration plus interface traction continuity and an elastic momentum residual constrain wall displacement. On a converging channel and a patient-like aorta, the PirateNets version reaches velocity relative errors of 0.0243 and 0.0372, below the 0.1066 and 0.1305 of a standard PINN of the same depth and width.

EDP-PI-DOSnet embeds classical energy-stable schemes into a physics-informed operator-splitting network so that the learned discrete operators satisfy a discrete energy dissipation law. The earlier PI-DOSnet did not guarantee that its learned linear operator was symmetric negative semidefinite, energy could rise between steps, and long-time extrapolation needed two or three rounds of retraining. On the Allen–Cahn equation, both EDP-PI-DOSnet variants keep the modified energy non-increasing at every observed time level and give smaller long-time prediction errors without retraining.

EDP-PI-DOSnet embeds classical energy-stable schemes into a physics-informed operator-splitting network so that the learned discrete operators satisfy a discrete energy dissipation law. The earlier PI-DOSnet did not guarantee that its learned linear operator was symmetric negative semidefinite, energy could rise between steps, and long-time extrapolation needed two or three rounds of retraining. On the Allen–Cahn equation, both EDP-PI-DOSnet variants keep the modified energy non-increasing at every observed time level and give smaller long-time prediction errors without retraining.

Physics-informed neural networks could previously show only small residuals at sampled points, not how large the error is over the continuous domain. This pipeline has a network search the Babenko equation in conformal coordinates for high-amplitude gravity–capillary waves, and staged neural corrections, Fourier projection and Newton polishing cut the maximum sampled residual by about seven orders of magnitude. The candidate is then frozen as an explicit trigonometric polynomial, interval arithmetic yields a rigorous uniform bound on the continuous residual, and Lean machine-checks the deductions from that numerical bound to the final residual theorem.

Physics-informed neural networks could previously show only small residuals at sampled points, not how large the error is over the continuous domain. This pipeline has a network search the Babenko equation in conformal coordinates for high-amplitude gravity–capillary waves, and staged neural corrections, Fourier projection and Newton polishing cut the maximum sampled residual by about seven orders of magnitude. The candidate is then frozen as an explicit trigonometric polynomial, interval arithmetic yields a rigorous uniform bound on the continuous residual, and Lean machine-checks the deductions from that numerical bound to the final residual theorem.

In power-system dynamic simulation, physics-informed neural networks built on multilayer perceptrons are limited by spectral bias toward smooth functions and struggle to capture electromechanical oscillations, while B-spline physics-informed Kolmogorov-Arnold networks are costly to train. mac-PIKAN embeds the closed-form modal response of the linearized swing equation as a fixed term in the trial solution and uses a Chebyshev-basis KAN to learn a nonlinear correction that vanishes at the initial time, so both initial rotor angle and speed hold exactly. On the single-machine infinite-bus benchmark its mean rotor-angle and frequency errors are 0.29% and 0.99%, against 1.59% and 5.26% for the PINN, and the advantage persists with less training data and at lower damping.

In power-system dynamic simulation, physics-informed neural networks built on multilayer perceptrons are limited by spectral bias toward smooth functions and struggle to capture electromechanical oscillations, while B-spline physics-informed Kolmogorov-Arnold networks are costly to train. mac-PIKAN embeds the closed-form modal response of the linearized swing equation as a fixed term in the trial solution and uses a Chebyshev-basis KAN to learn a nonlinear correction that vanishes at the initial time, so both initial rotor angle and speed hold exactly. On the single-machine infinite-bus benchmark its mean rotor-angle and frequency errors are 0.29% and 0.99%, against 1.59% and 5.26% for the PINN, and the advantage persists with less training data and at lower damping.

With only 20 to 50 coordinate–state observations, physics-informed neural networks (PINNs, which write the governing equations and initial and boundary conditions into the loss) reach the highest mean primary-target accuracy on a heat equation and three heterogeneous two-phase-flow benchmarks, and the water-saturation advantage holds through 1000 labels. Hierarchical loss weighting (H-PINN) and its gradient-direction-aware extension (HO-PINN) activate physical constraints as higher-priority losses approach prescribed scales, reducing run-to-run variability in selected settings. The cost is fitting time substantially higher than tuned gradient boosting, support-vector regression and Gaussian-process regression.

With only 20 to 50 coordinate–state observations, physics-informed neural networks (PINNs, which write the governing equations and initial and boundary conditions into the loss) reach the highest mean primary-target accuracy on a heat equation and three heterogeneous two-phase-flow benchmarks, and the water-saturation advantage holds through 1000 labels. Hierarchical loss weighting (H-PINN) and its gradient-direction-aware extension (HO-PINN) activate physical constraints as higher-priority losses approach prescribed scales, reducing run-to-run variability in selected settings. The cost is fitting time substantially higher than tuned gradient boosting, support-vector regression and Gaussian-process regression.

Physics-informed neural networks (PINNs) often lose accuracy on PDEs with sharp gradients, partly because the network struggles to represent high-frequency variation. CTPINN adds a learnable coordinate transformation alongside the solution network, mapping the physical domain to a computational domain where the solution is approximately affine and gradients are more evenly distributed. The explicit construction restores accuracy on boundary-layer problems, and the implicit construction, plugged into PirateNet, SPINN and QRPINN, improves accuracy across several benchmark equations.

Physics-informed neural networks (PINNs) often lose accuracy on PDEs with sharp gradients, partly because the network struggles to represent high-frequency variation. CTPINN adds a learnable coordinate transformation alongside the solution network, mapping the physical domain to a computational domain where the solution is approximately affine and gradients are more evenly distributed. The explicit construction restores accuracy on boundary-layer problems, and the implicit construction, plugged into PirateNet, SPINN and QRPINN, improves accuracy across several benchmark equations.

On the Heston stochastic-volatility pricing equation, BeTINN extends time-induced modulation to joint conditioning on time and model parameters, adding a residual pathway that processes the full input directly. Under capacity-matched comparisons it reaches the lowest mean absolute error on normalized option value, first-order sensitivities and second-order curvature, while validation physics loss only partly reflects the accuracy of those derivative quantities.

On the Heston stochastic-volatility pricing equation, BeTINN extends time-induced modulation to joint conditioning on time and model parameters, adding a residual pathway that processes the full input directly. Under capacity-matched comparisons it reaches the lowest mean absolute error on normalized option value, first-order sensitivities and second-order curvature, while validation physics loss only partly reflects the accuracy of those derivative quantities.

Effective thermal conductivity of porous silicon has long scattered because pore morphology and substrate doping differ, and no single model covered both. Gas-microphone photoacoustic measurements, atomistic molecular dynamics and physics-informed neural networks were compared on the same 48% to 75% porosity layers: molecular dynamics matched heavily doped columnar structures, while the networks matched the stochastic spongy networks of lightly and moderately doped silicon. Symbolic regression then combined porosity and temperature into one analytical expression, with mean absolute percentage error no more than 6.5% on the training set over 5% to 85% porosity and 300 to 900 K.

Effective thermal conductivity of porous silicon has long scattered because pore morphology and substrate doping differ, and no single model covered both. Gas-microphone photoacoustic measurements, atomistic molecular dynamics and physics-informed neural networks were compared on the same 48% to 75% porosity layers: molecular dynamics matched heavily doped columnar structures, while the networks matched the stochastic spongy networks of lightly and moderately doped silicon. Symbolic regression then combined porosity and temperature into one analytical expression, with mean absolute percentage error no more than 6.5% on the training set over 5% to 85% porosity and 300 to 900 K.