# Physics-Informed Neural Network

For formulation discovery with few measurements and a fixed experimental budget, KolTa splits property prediction into component base contributions, role-conditioned interactions and an anchored temperature response, trained with a derivative-based physics constraint. Its acquisition rule calibrates ensemble disagreement together with the physics residual at the target temperature against out-of-fold errors. Across binary-mixture viscosity, drug solubility, ionic-liquid viscosity and simulated electrolyte conductivity, mean AULC ranks first or second on every task, and on the drug task the median queries to reach 80% recovery are about 18% fewer than the fastest comparator.

For formulation discovery with few measurements and a fixed experimental budget, KolTa splits property prediction into component base contributions, role-conditioned interactions and an anchored temperature response, trained with a derivative-based physics constraint. Its acquisition rule calibrates ensemble disagreement together with the physics residual at the target temperature against out-of-fold errors. Across binary-mixture viscosity, drug solubility, ionic-liquid viscosity and simulated electrolyte conductivity, mean AULC ranks first or second on every task, and on the drug task the median queries to reach 80% recovery are about 18% fewer than the fastest comparator.

Cloth simulators usually cut motion into fixed timesteps and return states only on the solver grid, so querying arbitrary times or derivatives means re-running or finite differences. ContinuousCloth represents the cloth trajectory as a differentiable function of time, trained under the residual of a spring-mass ordinary differential equation, and answers position and derivative queries at any timestamp. Across six boundary-condition scenarios its trajectory accuracy matches a high-precision fourth-order Runge-Kutta reference while dense-time evaluation runs up to 17.4 times faster.

Cloth simulators usually cut motion into fixed timesteps and return states only on the solver grid, so querying arbitrary times or derivatives means re-running or finite differences. ContinuousCloth represents the cloth trajectory as a differentiable function of time, trained under the residual of a spring-mass ordinary differential equation, and answers position and derivative queries at any timestamp. Across six boundary-condition scenarios its trajectory accuracy matches a high-precision fourth-order Runge-Kutta reference while dense-time evaluation runs up to 17.4 times faster.

Soliton solutions of the KdV equation normally require retraining a physics-informed neural network whenever the dispersion parameter changes, and long-time extrapolation tends to drift. The scaled TW-PINN first normalizes the equation, trains once in normalized coordinates, and inverse-scales back to the original variables; the one-soliton solver uses a traveling-wave coordinate, and the two-soliton case trains in two steps, learning individual speeds first and the interaction second. In numerical experiments the two-soliton model preserved profiles ten times beyond the training domain, and a three-soliton experiment reproduced collision and separation.

Soliton solutions of the KdV equation normally require retraining a physics-informed neural network whenever the dispersion parameter changes, and long-time extrapolation tends to drift. The scaled TW-PINN first normalizes the equation, trains once in normalized coordinates, and inverse-scales back to the original variables; the one-soliton solver uses a traveling-wave coordinate, and the two-soliton case trains in two steps, learning individual speeds first and the interaction second. In numerical experiments the two-soliton model preserved profiles ten times beyond the training domain, and a three-soliton experiment reproduced collision and separation.

Conservation laws for Newtonian systems usually require writing the equations of motion and then solving symmetry partial differential equations that depend on the potential. Here the dynamics is embedded through the Eisenhart-Duval lift into an extended metric, so the symmetry conditions become Killing or conformal Killing equations of that metric, solved by a Hamiltonian network that learns the potential from phase-space data plus a random-feature network. For the one-dimensional Ermakov-Pinney equation, the three Killing vectors of the flat extended plane and their nonlocal conservation laws are recovered; for the one-dimensional harmonic oscillator, the ten-dimensional conformal algebra is reconstructed under noise.

Conservation laws for Newtonian systems usually require writing the equations of motion and then solving symmetry partial differential equations that depend on the potential. Here the dynamics is embedded through the Eisenhart-Duval lift into an extended metric, so the symmetry conditions become Killing or conformal Killing equations of that metric, solved by a Hamiltonian network that learns the potential from phase-space data plus a random-feature network. For the one-dimensional Ermakov-Pinney equation, the three Killing vectors of the flat extended plane and their nonlocal conservation laws are recovered; for the one-dimensional harmonic oscillator, the ten-dimensional conformal algebra is reconstructed under noise.

Indoor heating usually relies on one or a few fixed-point temperature readings, which miss the spatial variation around an occupant. A physics-informed neural network drops time as an explicit input and instead infers how the temperature field evolves from spatial coordinates plus a few sensor readings and their rates of change; across nine horizontal planes in three room-scale configurations it matched a validated COMSOL multiphysics simulation with a maximum relative error below 1.85%. Wired into occupant-centric heating control, it showed an average numerical error of about 0.67% against COMSOL-based offline control, with inference fast enough for real-time use.

Indoor heating usually relies on one or a few fixed-point temperature readings, which miss the spatial variation around an occupant. A physics-informed neural network drops time as an explicit input and instead infers how the temperature field evolves from spatial coordinates plus a few sensor readings and their rates of change; across nine horizontal planes in three room-scale configurations it matched a validated COMSOL multiphysics simulation with a maximum relative error below 1.85%. Wired into occupant-centric heating control, it showed an average numerical error of about 0.67% against COMSOL-based offline control, with inference fast enough for real-time use.

A neural collocation scheme for semilinear time-fractional reaction–diffusion equations rewrites the Caputo memory term as cubic-Hermite product integration, using the network state and its autodifferentiated time slope as endpoint data on each time panel while the weakly singular weights are precomputed once before training. On the scalar benchmark v(t)=t⁴.2 the Hermite memory approximation is three to six orders of magnitude more accurate than the classical L1 formula on the tested meshes, and for a manufactured problem with α=0.7 a small CPU-trained network reaches a relative L2 error of 1.98×10⁻⁵ and a maximum error of 3.80×10⁻⁵.

A neural collocation scheme for semilinear time-fractional reaction–diffusion equations rewrites the Caputo memory term as cubic-Hermite product integration, using the network state and its autodifferentiated time slope as endpoint data on each time panel while the weakly singular weights are precomputed once before training. On the scalar benchmark v(t)=t⁴.2 the Hermite memory approximation is three to six orders of magnitude more accurate than the classical L1 formula on the tested meshes, and for a manufactured problem with α=0.7 a small CPU-trained network reaches a relative L2 error of 1.98×10⁻⁵ and a maximum error of 3.80×10⁻⁵.

Task allocation for collaborative robotic welding of dissimilar steel joints no longer divides work by robot-to-seam distance alone. The framework treats each robot as a player in an α-potential game and uses a physics-informed neural-network surrogate for repeated thermo-mechanical evaluation, so allocation accounts for residual stress and carbon-depleted-zone width. Across computational cases with 50 tasks and 2 robots, 100 tasks and 3 robots, and 200 tasks and 4 robots, it reduced normalized makespan and energy relative to distance-driven allocation while keeping lower predicted residual stress and carbon-depleted-zone width; these are computational predictions requiring independent experimental validation.

Task allocation for collaborative robotic welding of dissimilar steel joints no longer divides work by robot-to-seam distance alone. The framework treats each robot as a player in an α-potential game and uses a physics-informed neural-network surrogate for repeated thermo-mechanical evaluation, so allocation accounts for residual stress and carbon-depleted-zone width. Across computational cases with 50 tasks and 2 robots, 100 tasks and 3 robots, and 200 tasks and 4 robots, it reduced normalized makespan and energy relative to distance-driven allocation while keeping lower predicted residual stress and carbon-depleted-zone width; these are computational predictions requiring independent experimental validation.

Pricing American options on several assets is hard because the early-exercise boundary is unknown, and extending knot methods to d assets by tensor product needs O(N^d) knots. A physics-informed neural network represents the option value with a tanh feed-forward network, uses automatic differentiation for the multi-asset Black–Scholes operator, and trains on sampled collocation points instead of a grid. On a two-asset American basket put, the 8000-epoch configuration with a one-sided residual penalty and a smooth Fischer–Burmeister complementarity function gave 7.8669, an overestimate of 0.97, about 14%, against the Longstaff–Schwartz Monte Carlo reference of 6.9005, and the closest of eight training setups.

Pricing American options on several assets is hard because the early-exercise boundary is unknown, and extending knot methods to d assets by tensor product needs O(N^d) knots. A physics-informed neural network represents the option value with a tanh feed-forward network, uses automatic differentiation for the multi-asset Black–Scholes operator, and trains on sampled collocation points instead of a grid. On a two-asset American basket put, the 8000-epoch configuration with a one-sided residual penalty and a smooth Fischer–Burmeister complementarity function gave 7.8669, an overestimate of 0.97, about 14%, against the Longstaff–Schwartz Monte Carlo reference of 6.9005, and the closest of eight training setups.