Physics-Informed Neural Networks: PINN Tutorial with Drug Decay Example

PINNs demonstrate remarkable versatility in solving diverse scientific computing problems. For forward problems, PINNs can accurately reconstruct solutions throughout entire domains using only boundary and initial condition data, as demonstrated on challenging problems like the Allen-Cahn equation where traditional numerical methods suffer from oscillations. For inverse problems, PINNs treat unknown physical parameters (such as λ₁ and λ₂ in reaction-diffusion equations) as trainable variables within the neural network architecture. By optimizing the combined loss function that simultaneously minimizes data misfit and PDE residuals, PINNs can infer governing equation parameters from interior domain measurements without requiring explicit boundary or initial condition information.

PINN solves two problem types: data-driven solutions (known PDE coefficients, find solution function u) and data-driven discovery (known solution form, find coefficients λ). The general PDE form is N(u)=0, where u is the solution function and N is a nonlinear operator. The PINN loss combines data loss (fitting predictions to known data including initial/boundary conditions) and physics loss (minimizing PDE residual). The DeepXDE library simplifies implementation by handling complex geometries through set operations on basic shapes. PINN's unique ability to learn both high and low frequency components simultaneously improves learning speed and stability. Beyond forward problems, PINN solves inverse problems by treating PDE coefficients as trainable parameters, enabling discovery of governing equations from data—a capability traditional numerical methods cannot provide.

PINNs fail for the inviscid Burgers equation (ν → 0) because shocks form and gradients blow up like 1/√ν, violating coercivity conditions. However, PINNs work for viscous Burgers equation (ν > 0) where gradients remain bounded. For inviscid cases, weak formulations with neural network test functions and entropy conditions restore effectiveness, reducing error from 20-23% to about 2%. PINNs excel for inverse problems where data is available only in the interior. The method augments PDE residuals with data residuals at measurement points. For parabolic PDEs, Carleman estimates provide coercivity. PINNs solve 100-dimensional inverse problems to 2% error and reconstruct Stokes equation velocity and pressure from interior measurements with 3.5% error for pressure.

For inverse problems, PINNs treat unknown parameters of the differential equation as learnable parameters alongside the neural network weights. The boundary loss is replaced with a data loss that matches observations at arbitrary points in the domain. During training, both the neural network (solution) and the differential equation parameters are optimized simultaneously, effectively solving forward simulation and inverse problem together.

Inverse problems involve determining unknown parameters or boundary conditions from partial measurements of the solution. PINNs solve these by combining the PDE residual (enforcing physics everywhere) with a data fidelity term (matching measurements at known locations). The neural network learns to extend measurements from accessible regions to the entire domain while respecting the governing physics. This approach works effectively even with noisy measurements and demonstrates robustness to uncertainty.

Several advanced strategies improve PINN training stability and efficiency: finite difference approximations for gradient computation offer modest speedups over automatic differentiation; hard constraints encode boundary conditions directly into the solution ansatz, eliminating competition between loss terms; self-adaptive weighting treats loss coefficients as learnable parameters, concentrating training effort on high-residual regions; adaptive sampling dynamically concentrates collocation points where the PDE residual is largest, mimicking adaptive mesh refinement. These methods transform difficult optimization problems into more tractable forms, though they introduce additional complexity in implementation and hyperparameter management.

PINN performance critically depends on optimizer selection and learning rate scheduling. Standard optimizers like AdamW may get stuck in local minima, while advanced optimizers incorporating Hessian information can drive error to machine precision. One-cycle learning rates address convergence difficulties by increasing then decreasing the learning rate during training. Quasi-Monte Carlo methods improve high-dimensional sampling efficiency compared to standard Monte Carlo. These practical considerations demonstrate that successful PINN deployment requires careful attention to optimization strategy alongside proper physics encoding.

At each colocation point, automatic differentiation measures how the predicted temperature changes with time and curves across space. Whatever fails to cancel is the residual. The physics loss squares and averages these residuals. The data loss says 'Obey the temperatures we know at the edges and at the start,' while the physics loss says 'Obey the heat equation everywhere inside.' Boundary and initial data describe the particular situation we set up, while the differential equation describes the physical law. Deliberately balancing the loss terms is crucial because one loss can dominate and the model will satisfy the loud one while ignoring the quiet one.

A PINN loss function typically contains three components: (1) Boundary loss terms that enforce initial/boundary conditions by ensuring the network output matches known values at specific points; (2) Physics loss terms that enforce the governing differential equation across the domain; (3) Optional data loss terms for supervised learning when observational data is available. Hyperparameters (lambdas) balance these terms to prevent any single component from dominating the optimization.

PINNs demonstrate surprising fragility when applied to seemingly simple physical problems. Research on advection equations (ut + βux = 0), reaction equations (ut - ρu(1-u) = 0), and reaction-diffusion systems reveals consistent failure modes. For advection equations, PINNs fail when velocity coefficients exceed certain thresholds despite analytical solutions existing. For reaction equations, standard PINN training produces completely incorrect predictions. Loss landscape analysis reveals that as PDE residual weights increase, the optimization landscape becomes increasingly complex and rugged. This creates a fundamental trade-off: appropriate weights enable physically accurate solutions but make optimization extremely difficult; very small weights allow easy optimization but produce unphysical results.

Physics-Informed Neural Networks (PINNs) combine data-driven neural networks with mathematical models based on physical laws (ordinary differential equations) to solve complex systems like epidemiological models; in this application, PINNs were used to identify key transmission parameters (beta and kappa) for a six-compartment SBIHDM model of COVID-19 dynamics, achieving better prediction accuracy than purely data-driven LSTM approaches by incorporating the system of ODEs into the neural network's loss function alongside data fitting.

PINNs enable realistic applications including: (1) Epidemiological modeling combining SEIR models with neural networks to predict disease spread using human mobility data, enabling sensitivity analysis to evaluate policy interventions; (2) Medical imaging applications reconstructing blood flow from noisy 4D flow MRI data while satisfying Navier-Stokes equations and inferring arterial wall geometry and wall shear stresses. These applications demonstrate synthesizing observational data with physical laws for practical scientific and engineering challenges.

UPIN can be applied to any physical system with unknown components and experimental data, including quantitative systems pharmacology. The method identifies unknown components of differential equations from noisy or sparse data, aiding physical discovery and helping determine unknown drug actions or pathway components when they are not known a priori. This represents a powerful tool for pharmaceutical research where the exact mechanism of drug action may be unknown, allowing researchers to use experimental data to determine the functional form of unknown components in biological systems.

PINNs have significant real-world applications when resources are limited but physical understanding exists. A compelling example is computational microscopy, where researchers use PINNs to enhance low-quality microscope images by incorporating physical knowledge of light propagation and imaging physics. When expensive high-resolution microscopes are unavailable, PINNs can produce image quality comparable to ground-truth measurements by solving inverse problems constrained by physical laws. This demonstrates how PINNs bridge the gap between theoretical physics and practical engineering challenges, enabling valuable research outcomes even with resource constraints.

UPINNs were applied to model chemotherapy drug action on tumor cells, including tumor cell count (n) and chemotherapy concentration (C) over time. The drug action function G(n) describes how the drug affects cell growth. Different drug action models (log kill, Norton Simon, Emax) were tested. The goal was to identify which drug action model best describes the data rather than assuming a specific model. UPINNs could identify the correct drug action function from data: linear for log kill, quadratic for Norton Simon, and asymptotic for Emax. The method performed well on both noiseless and noisy data, demonstrating its ability to discover biological mechanisms from experimental data.