Universal Physics-Informed Neural Networks

• Universal PINNs are physics-constrained neural networks that integrate governing laws with observational data to solve complex differential equations.
• They employ hybrid formulations that optimize both data matching and physics residuals, enabling flexible adaptation across diverse scientific domains.
• Applications span from pharmacodynamics to celestial mechanics, showcasing their capability to infer hidden dynamics and improve physical fidelity.

Universal Physics-Informed Neural Network (UPINN) denotes, in current usage, not a single canonical neural architecture but a family of physics-constrained learning formulations whose common aim is to solve, infer, or discover differential-equation structure within one optimization framework. In the broad scientific-machine-learning view, the term aligns with a unifying philosophy in which neural networks are trained against governing laws, boundary or initial conditions, and data simultaneously, rather than through a purely data-driven fit or a purely mesh-based solver (Raissi et al., 2024). In a narrower and increasingly common sense, a UPINN is a PINN-style model for partially known dynamics, typically written as $\frac{du}{dt}=F(u,t)+G(u,t)$, where the known component $F$ is retained explicitly and the hidden term $G$ is represented by a trainable neural network that is learned jointly with the solution surrogate (Podina et al., 2024, Podina et al., 2024).

1. Conceptual scope and the meaning of “universal”

The review literature presents PINNs as a central paradigm in scientific machine learning because they fuse data with governing physical laws in a single neural-network framework. The core idea is that a neural network serves as a trial function for an unknown state, while automatic differentiation supplies the derivatives needed to evaluate the residual of the governing ODE, PDE, or stochastic equation. This immediately places forward problems, inverse problems, and mixed settings on the same footing: the governing law may be fully known, partially known, or only partially constrained by scattered observations (Raissi et al., 2024).

Within that landscape, “universal” has acquired at least two technically distinct meanings. One is architectural and methodological: the same optimization pattern can be reused across multiple classes of physical systems. The other is inferential: the model can discover hidden terms, constitutive components, or missing physics rather than only estimating scalar coefficients in a fixed equation. The variational-physics literature makes this distinction explicit by describing a “universal computational workflow” in which one encodes physics as a scalar objective, differentiates automatically, and optimizes iteratively; it does not claim a universal physical action functional or a theorem that covers all systems (Razdan et al., 3 Jan 2026).

This distinction matters because UPINNs are often contrasted with both classical numerical solvers and ordinary neural networks. Classical solvers remain accurate but depend on discretization, meshes, and explicit governing equations; pure neural networks fit data but do not automatically respect physical constraints. UPINNs occupy the intermediate regime emphasized in the PINN review: part of the physics is known, part is unknown, and the learning problem is to recover the state together with the missing structure (Raissi et al., 2024).

2. Mathematical formulation

The most direct UPINN formulation in the discovery literature assumes a partially known differential equation,

$\frac{du}{dt}=F(u,t)+G(u,t),$

where $F(u,t)$ is known and $G(u,t)$ is an unknown hidden term. A UPINN then learns two coupled objects: a surrogate solution $U_{NN}(t)$ approximating the trajectory and a neural representation $G_{NN}$ of the hidden term. The training objective is written as a sum of a data-matching term and a differential-equation residual term, so that the learned trajectory must both fit observations and satisfy the physics-constrained residual under automatic differentiation (Podina et al., 2024, Podina et al., 2024).

This structure generalizes the standard PINN formulation in which a neural network approximation $u_\theta$ is trained so that the residual $\mathcal{N}[u_\theta]$ is small at collocation points while boundary, initial, and measurement constraints are also enforced. In the broader PINN literature, the trainable variables may include only network weights, or both network weights and unknown physical parameters. The variational-physics treatment states this directly as $F$0 with optimization over

$F$1

and interprets PINNs as a computational bridge between classical variational principles and machine-learning optimization (Razdan et al., 3 Jan 2026).

The same formal pattern appears in more specialized domains. In Bayesian PINNs for nonlinear dynamical systems, the loss is decomposed into data and physics contributions, weighted either by a fixed scalar $F$2 or by a learned time-dependent $F$3; in addition, the unknown physical parameters can be included in the trainable set $F$4 and subsequently embedded in a Bayesian posterior (Linka et al., 2022). In perturbative QCD, the unknowns are not simple trajectory components but fragmentation functions $F$5, and the physics residual is supplied by the time-like DGLAP evolution equations rather than a low-dimensional ODE (Dai et al., 28 Jan 2026).

3. Architectural patterns and training mechanics

A defining property of the UPINN literature is that “universal” does not imply one fixed network template. The review of PINNs and extensions explicitly frames universality as a family of design principles: adaptive loss weighting, domain decomposition, causal training for long-time integration, stochastic and fractional generalizations, and hybrid variational formulations all preserve the central idea of embedding physics into learning while altering the optimizer, the decomposition strategy, or the form of the residual (Raissi et al., 2024).

Loss balancing is a recurrent issue. Standard PINNs use weighted sums of data, residual, boundary, and initial-condition terms, but fixed weights are often hard to tune. The review identifies Neural Tangent Kernel-based calibration, trainable per-collocation-point weights cast as a min-max problem, and residual-based adaptive variants as responses to the common pathology that different regions of the domain train at very different rates. The same theme appears in Bayesian PINNs, where performance depends sensitively on the data-versus-physics balance: small $F$6 makes the model close to a classical NN, large $F$7 makes it close to the physics model alone, and a self-adaptive variant learns $F$8 directly from the data (Linka et al., 2022).

Application-specific UPINNs often use multiple subnetworks. In chemotherapy pharmacodynamics, the implementation is fully connected, with 8 hidden layers, 20 hidden units per layer, tanh activations, 5000 Adam iterations followed by L-BFGS until convergence, equal weighting of the MSE and PINN residual, and inputs scaled to $F$9 (Podina et al., 2024). In the fragmentation-function study, the architecture is more structured: a cascaded two-network design in Mellin space, where NN1 takes $G$0 and outputs a latent feature, NN2 takes the concatenation of $G$1 and that latent feature, positivity is enforced through $G$2, and the endpoint condition $G$3 is built into the representation through

$G$4

That system is trained with a four-term loss whose weights are chosen as $G$5, reflecting the need to enforce DGLAP evolution and fit Mellin-transformed data simultaneously (Dai et al., 28 Jan 2026).

4. Hidden-term discovery and dimensional analysis

A particularly explicit UPINN discovery pipeline appears in the work combining dimensional analysis, UPINNs, and symbolic regression. There the objective is not to recover an entire equation from scratch but to identify a hidden term more effectively when the rest of the differential equation is known. The workflow is: write the dynamics as a sum of known and unknown parts, use Ipsen’s method to construct dimensionless variables, train the UPINN on the nondimensionalized system, and then pass the learned hidden term to AI Feynman for symbolic recovery (Podina et al., 2024).

The extended logistic-growth example illustrates the reduction in complexity. Starting from

$G$6

the hidden term is

$G$7

Using the dimensionless variables

$G$8

the transformed equation becomes

$G$9

When the hidden term is treated as unknown, it reduces to a one-variable function $\frac{du}{dt}=F(u,t)+G(u,t),$0, and the reported recovered term is

$\frac{du}{dt}=F(u,t)+G(u,t),$1

The rotating-bead example follows the same logic: a second-order equation is decomposed into known and hidden parts, dimensionless variables are chosen so that the hidden term depends on fewer arguments, and the symbolic-regression step operates on the reduced representation (Podina et al., 2024).

The significance of this pipeline is methodological rather than merely cosmetic. The paper argues that nondimensionalization reduces the number of hidden-term inputs, normalizes scales, shrinks the symbolic-regression search space, and guarantees dimensional consistency of the recovered expression. In the same study, the broader symbolic-regression experiments report large runtime reductions after nondimensionalization, such as free fall from 1839 s to 486 s and gravitational force from 2618 s to 128 s, and these results are used to motivate dimensional analysis as a structural improvement to the entire UPINN discovery process (Podina et al., 2024).

5. Representative application domains

In quantitative systems pharmacology, UPINNs are used to learn unknown chemotherapy drug-action terms directly from synthetic or biological time-series data. The base model is

$\frac{du}{dt}=F(u,t)+G(u,t),$2

and the hidden pharmacodynamic law $\frac{du}{dt}=F(u,t)+G(u,t),$3 is learned for three standard forms: log-kill, Norton-Simon, and $\frac{du}{dt}=F(u,t)+G(u,t),$4. For equally spaced noiseless data, the reported MSEs between true and learned $\frac{du}{dt}=F(u,t)+G(u,t),$5 are $\frac{du}{dt}=F(u,t)+G(u,t),$6 for log-kill, $\frac{du}{dt}=F(u,t)+G(u,t),$7 for Norton-Simon, and $\frac{du}{dt}=F(u,t)+G(u,t),$8 for $\frac{du}{dt}=F(u,t)+G(u,t),$9. The same paper also uses a single UPINN to fit multiple dosage datasets simultaneously, learning dose-dependent functions $F(u,t)$0 and $F(u,t)$1, and applies the method to doxorubicin-treated triple-negative breast-cancer cell lines, where the learned $F(u,t)$2 exhibits a plateau followed by increased killing and is stated not to be representable by the simple McKenna forms alone or even their weighted average (Podina et al., 2024).

For nonlinear dynamical systems with uncertainty quantification, Bayesian PINNs extend the same data-plus-physics template by introducing priors on the neural and physical parameters. In the damped-oscillator surrogate used for seasonal COVID-19 outbreak dynamics, the residual is

$F(u,t)$3

and the method is used in both forward and inverse modes. The reported parameter recovery includes $F(u,t)$4 and $F(u,t)$5 for projectile motion with linear drag, while the learned self-adaptive weight is reported as $F(u,t)$6 (Linka et al., 2022).

In celestial mechanics, PINNs are used as physics-regularized surrogates for the three-body problem. The governing equations are Newtonian gravitational ODEs for three point masses, and the training loss augments trajectory error with a weighted physics term. The principal empirical result is a dramatic reduction in the physics-informed residual: the mean-squared physics-informed error is reported as $F(u,t)$7 for the baseline DNN and $F(u,t)$8 for the PI DNN, with the PI ResNet reaching $F(u,t)$9. At the same time, the standard ResNet attains better raw MAE and RMSE than the PI variant, making the main advantage of the PINN physical fidelity rather than uniformly superior pointwise prediction (Pereira et al., 6 Mar 2025).

In high-energy QCD phenomenology, a universal PINN framework is used to extract fragmentation functions without assuming a fixed parametric form at an input scale. The neural representation is trained directly against SIA data while satisfying time-like DGLAP evolution in Mellin space. Universality is then assessed by transfer across processes: fragmentation functions extracted only from $G(u,t)$0 data are used to compute charged-hadron spectra in $G(u,t)$1 and $G(u,t)$2 collisions at $G(u,t)$3. The paper reports agreement with a direct RK4 DGLAP evolution solver within about 5% at $G(u,t)$4 GeV, a closure-test maximum deviation of about 20% in $G(u,t)$5, and good agreement with collider data across the quoted energy range (Dai et al., 28 Jan 2026).

6. Limitations, interpretation, and adjacent paradigms

The principal misconception surrounding UPINNs is that “universal” means a single architecture that solves arbitrary physics problems without reformulation. The cited literature does not support that reading. The variational-physics treatment is explicit that universality is computational rather than ontological: a single optimization pattern can span optics, mechanics, heat conduction, eigenvalue problems, and PINNs, but not every physical system has a true action principle, and dissipative systems often require residual-based PINNs rather than classical variational minimization (Razdan et al., 3 Jan 2026).

Methodological limitations are equally recurrent. The chemotherapy study highlights the absence of uncertainty quantification, identifiability issues when multiple hidden functions are learned simultaneously, sensitivity of adjusted sampling under noise, and limited extrapolation beyond the observed range of $G(u,t)$6 or dosage (Podina et al., 2024). The Bayesian PINN study emphasizes sensitivity to the loss-weighting parameter $G(u,t)$7, instability of self-adaptive schemes under poor scaling or preconditioning, and the substantially higher computational cost of BPINNs (Linka et al., 2022). The three-body study adds optimization instability, exploding gradients, close-encounter singular behavior, and a trade-off between physical coherence and pure prediction metrics over finite horizons (Pereira et al., 6 Mar 2025). The broader review literature also states that rigorous convergence results are available only for certain linear elliptic and parabolic PDEs, with stronger convergence when boundary or initial conditions are satisfied exactly (Raissi et al., 2024).

A second misconception is to collapse all “universal physics” models into the PINN category. The diffusion-based “Universal Physics Simulation” model is explicitly not a UPINN in the standard sense: it is a sketch-guided diffusion transformer trained with diffusion denoising, reconstruction, edge, LPIPS, and prior-alignment losses, and it contains no PDE residual term. Its validation is confined to 2D FDTD electromagnetics, and the paper itself admits that the proposed multi-domain universality remains a claim rather than an experimental demonstration (Camburn, 13 Jul 2025).

Taken together, these works define UPINNs less as a finished architecture than as a research program. Their common premise is that physical structure can be embedded directly into learnable surrogates, but the concrete form of that embedding varies substantially with the problem class: hidden-term inference in partially known ODEs, adaptive or decomposed PINN training for difficult PDEs, Bayesian uncertainty quantification for nonlinear dynamics, and integro-differential enforcement of QCD evolution. The term “universal” therefore refers most defensibly to breadth of computational scope and transferability of the physics-informed optimization pattern, not to a universal theorem, a unique network topology, or a domain-independent guarantee of performance.