PINNSR-DA: Physics-Informed Sparse Regression

• PINNSR-DA is a physics-informed neural network paradigm that couples sparse regression with dimensional analysis and data assimilation for equation discovery.
• It reconstructs latent state trajectories by integrating noisy measurements with differentiable physics constraints using composite loss functions and adaptive sampling.
• Its versatile applications in granular rheology, orbital mechanics, power-system dynamics, and turbulence highlight its practical value and interpretability.

PINNSR-DA denotes a physics-informed neural-network paradigm in which data-driven reconstruction or equation discovery is constrained by governing physics and coupled to an additional mechanism such as sparse regression, data assimilation, dimensional analysis, adaptive sampling, or symbolic regression. In the most explicit usage, the term names a framework for discovering governing equations of sheared granular materials by combining physics-informed neural networks with sparse regression and machine learning-based dimensional analysis (Han et al., 18 Sep 2025). In related literature, the same label is used analogically for PINN-based sparse reconstruction via data assimilation in orbital mechanics, turbulence, and power-system dynamics, and for PINN workflows that refine collocation, augment inputs, or extract symbolic structure (Varey et al., 2024).

1. Terminology and scope

The literature summarized here does not use PINNSR-DA as a single fully standardized acronym. One source defines it explicitly as “physics-informed neural networks with sparse regression” combined with machine learning-based dimensional analysis for granular rheology discovery (Han et al., 18 Sep 2025). Other sources describe methods “akin to PINNSR-DA,” identify “a natural mapping” to the acronym, or present closely related PINN-based sparse reconstruction and data-assimilation workflows (Varey et al., 2024).

Usage in the literature	Role of the method	Representative paper
Explicit PINNSR-DA	Physics-informed neural networks with sparse regression plus dimensional analysis for governing-law discovery	(Han et al., 18 Sep 2025)
Akin to PINNSR-DA	State reconstruction via data assimilation with a learned anomalous acceleration in satellite SDA	(Varey et al., 2024)
PINN-based sparse reconstruction via DA	Mean-flow reconstruction from sparse measurements with RANS or RANS-SA constraints	(Patel et al., 2023)
Natural mapping to PINNSR-DA	Deep adaptive sampling for PINNs using KRnet to refine collocation	(Tang et al., 2021)
Related PINN extensions	Dimension augmentation and symbolic regression pipelines for accuracy or interpretability	(Guan et al., 2022, Majumdar et al., 2023)

A common denominator across these usages is the replacement of purely empirical fitting by a composite optimization in which a neural surrogate is trained jointly against measurements and physics. The unknown quantity may be a state trajectory, an anomalous forcing term, a coefficient vector in a sparse library, a turbulence correction, or a closed-form symbolic expression. This suggests that PINNSR-DA is best understood as a family resemblance term rather than a single canonical algorithm.

2. Recurrent methodological structure

Across the cited works, the core object is a neural approximation of a latent field or trajectory. In satellite estimation, the state is $\mathbf{x}(t)=[\mathbf{r}(t),\mathbf{v}(t)]$ and the acceleration is decomposed into known physics plus a learned anomalous term, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$ (Varey et al., 2024). In power-system identification, the network approximates bus-angle trajectories $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$, with automatic differentiation supplying $\dot{\mathbf{u}}(t)$ and $\ddot{\mathbf{u}}(t)$ to enforce the multi-machine swing equations (Stiasny et al., 2020). In turbulent-flow reconstruction, the outputs include mean velocity, pressure-related variables, corrective forcing, and, in the SA-augmented case, the working variable $\tilde{\nu}$ (Patel et al., 2023). In granular rheology discovery, the network represents latent responses $\tau(t)$ and $\gamma(t)$ while sparse regression identifies an active ODE structure (Han et al., 18 Sep 2025).

The loss is likewise composite. Satellite SR-DA blends observation misfit, an optional physics residual, and regularization on the learned anomalous acceleration, although the reported implementation minimizes the observation term while enforcing dynamics implicitly through a Neural ODE (Varey et al., 2024). Power-system PINNs minimize $\mathcal{L}_z+\mathcal{L}_c$, where $\mathcal{L}_z$ is the angle/frequency data loss and $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$0 is the collocation loss from the swing-equation residual (Stiasny et al., 2020). Granular PINNSR uses

$$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$1

with $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$2 a measurement loss, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$3 a physics-residual loss, and $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$4 the sparsity prior on the candidate library coefficients (Han et al., 18 Sep 2025). In turbulent PINN-DA-SA, the objective combines data, strong-form RANS residuals, the SA transport residual, boundary conditions, and an $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$5 penalty on the solenoidal corrective forcing (Patel et al., 2023).

A second recurring feature is differentiability of the physical solver. The satellite method uses TorchDiffEq so that automatic differentiation propagates through the ODE solver to neural parameters and, when fitted, initial conditions (Varey et al., 2024). The power-system and turbulence formulations compute strong-form residuals directly by automatic differentiation (Stiasny et al., 2020, Patel et al., 2023). This differentiable-physics coupling is the mechanism that distinguishes these workflows from purely black-box sequence models or standard regression pipelines.

3. Explicit PINNSR-DA in granular rheology

In its explicit formulation, PINNSR-DA is a framework for discovering physically interpretable, dimensionally consistent governing equations for sheared granular materials in steady and transient states (Han et al., 18 Sep 2025). The physical setting is three-dimensional discrete element method simulation of wall-confined oscillatory shear on dry, polydisperse granular assemblies, with periodicity in $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$6 and $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$7, variable height $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$8–$$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$9, rough top and bottom plates, disabled gravity, Hertz–Mindlin contacts, and a constant overburden pressure $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$0 applied at the top wall. Two loading protocols are used: sinusoidal shear and Heaviside rate reversal. The inertial number is kept in the quasi-static regime,

$\mathbf{u}(t)\approx\{\delta_k(t)\}_k$1

and the effective friction is defined by $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$2.

The methodology has two stages. First, PINNSR discovers a parsimonious ODE for stress evolution from noisy DEM signals. The starting hypothesis is

$\mathbf{u}(t)\approx\{\delta_k(t)\}_k$3

where $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$4 is a library of 23 candidate terms built from $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$5, their time derivatives, and products such as $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$6, $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$7, $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$8, $\mathbf{u}(t)\approx\{\delta_k(t)\}_k$9, $\dot{\mathbf{u}}(t)$0, $\dot{\mathbf{u}}(t)$1, and $\dot{\mathbf{u}}(t)$2 (Han et al., 18 Sep 2025). The DNN has 6 hidden layers with 20 tanh neurons per layer, takes as input $\dot{\mathbf{u}}(t)$3 and the identifier $\dot{\mathbf{u}}(t)$4, and outputs $\dot{\mathbf{u}}(t)$5 and $\dot{\mathbf{u}}(t)$6. Collocation density is set to $\dot{\mathbf{u}}(t)$7 with $\dot{\mathbf{u}}(t)$8. Optimization uses pretraining with L-BFGS on an $\dot{\mathbf{u}}(t)$9-relaxed objective, followed by alternating direction optimization: STRidge thresholding for $\ddot{\mathbf{u}}(t)$0 and Adam updates for $\ddot{\mathbf{u}}(t)$1. Residual-based adaptive collocation with KDE concentrates points where $\ddot{\mathbf{u}}(t)$2 is large, particularly near reversals.

Second, machine learning-based dimensional analysis learns dimensionless coefficient laws. For each active coefficient $\ddot{\mathbf{u}}(t)$3, the method assumes a dominant dimensionless number

$\ddot{\mathbf{u}}(t)$4

subject to $\ddot{\mathbf{u}}(t)$5, where $\ddot{\mathbf{u}}(t)$6 is the stated $\ddot{\mathbf{u}}(t)$7 dimension matrix for the ordered variables $\ddot{\mathbf{u}}(t)$8 (Han et al., 18 Sep 2025). A two-level learning alternates between grid search over the null-space coefficients $\ddot{\mathbf{u}}(t)$9 and polynomial fitting of $\tilde{\nu}$0, selecting the best dimensionless group by test-set $\tilde{\nu}$1.

Across sinusoidal and Heaviside cases, the framework repeatedly identifies the same three-term evolution law for the effective friction:

$\tilde{\nu}$2

For the representative configuration $\tilde{\nu}$3, $\tilde{\nu}$4, $\tilde{\nu}$5, $\tilde{\nu}$6, $\tilde{\nu}$7, and $\tilde{\nu}$8, the coefficients are $\tilde{\nu}$9, $\tau(t)$0, and $\tau(t)$1 (Han et al., 18 Sep 2025).

Term	Interpretation in the source	Mathematical form
Linear response	Direct, rate-driven change of friction from applied shear	$\tau(t)$2
Energy dissipation	Coupling of friction state to shear-rate intensity	$\tau(t)$3
Nonlinear response	Self-driven evolution of friction	$\tau(t)$4

Dimensional analysis yields the dominant groups

$\tau(t)$5

and, with $\tau(t)$6, the coefficient laws are simplified to

$\tau(t)$7

with

$\tau(t)$8

where $\tau(t)$9, $\gamma(t)$0, and $\gamma(t)$1. The reported test-set coefficients of determination are $\gamma(t)$2, $\gamma(t)$3, and $\gamma(t)$4 for $\gamma(t)$5, $\gamma(t)$6, and $\gamma(t)$7 (Han et al., 18 Sep 2025). Steady-state reduction gives a configuration-insensitive static friction, with global fit $\gamma(t)$8, and the transient relaxation time obeys

$\gamma(t)$9

The paper further supplies a microstructural interpretation through the fabric tensor $\mathcal{L}_z+\mathcal{L}_c$0. Since $\mathcal{L}_z+\mathcal{L}_c$1 is approximately linear in the dominant component $\mathcal{L}_z+\mathcal{L}_c$2, the discovered friction law matches the reduced fabric-evolution structure, linking the three ODE terms to shear-induced anisotropy, dissipative stabilization, and nonlinear modulation via fabric anisotropy (Han et al., 18 Sep 2025).

4. Data-assimilation and state-reconstruction instantiations

A closely related use of the PINNSR-DA idea appears in satellite state estimation for Space Domain Awareness. The task is to reconstruct an orbital state and a time-varying, low-thrust acceleration profile from SSN-like angles-only observations, specifically right ascension and declination from a single optical ground telescope (Varey et al., 2024). The state dynamics are written as

$\mathcal{L}_z+\mathcal{L}_c$3

with the learned anomalous term represented by a compact MLP. The reported implementation uses a simple MLP with two hidden layers of width 100, tanh activations, and output dimension 3. Training uses a 2-day fit window with 30 angles-only measurements, Gaussian measurement noise $\mathcal{L}_z+\mathcal{L}_c$4 arcsec per component, Adam with initial learning rate $\mathcal{L}_z+\mathcal{L}_c$5, step-wise decay every 100 epochs by a factor of $\mathcal{L}_z+\mathcal{L}_c$6, and total 20,000 epochs. Initial conditions are updated every $\mathcal{L}_z+\mathcal{L}_c$7 training iterations by a batch least-squares fit. For a GEO simulation with unmodeled acceleration on the order of $\mathcal{L}_z+\mathcal{L}_c$8, the PINN-coupled model reduces observation RMSE from $\mathcal{L}_z+\mathcal{L}_c$9 arcsec to $\mathcal{L}_z$0 arcsec and 5-day position error from $\mathcal{L}_z$1 km to $\mathcal{L}_z$2 km; the corresponding 5-day velocity error improves from $\mathcal{L}_z$3 m/s to $\mathcal{L}_z$4 m/s (Varey et al., 2024).

In power-system dynamics, PINNs are used for nonlinear system identification and dynamic assessment from PMU-like data. The formulation embeds the multi-machine swing equations,

$\mathcal{L}_z$5

within a collocation loss, while the network approximates bus-angle trajectories and automatic differentiation yields frequency and acceleration (Stiasny et al., 2020). The architecture uses $\mathcal{L}_z$6 hidden layers with 30 neurons per layer, tanh activation, identity output, measurement sampling every $\mathcal{L}_z$7 s over $\mathcal{L}_z$8 s, and 20 collocation points per measurement. In the 4-bus test system, PINN achieves relative errors below 1% for System A and some parameters in System B, remains reasonably accurate in low-inertia fast dynamics where UKF struggles, and is robust to Gaussian or uniform noise up to 5%, although System C exhibits slow convergence due to a flat optimization landscape (Stiasny et al., 2020).

For turbulent mean-flow reconstruction, a direct PINN-based sparse reconstruction via data assimilation is presented through PINN-DA-Baseline and PINN-DA-SA. The method combines sparse mean-velocity measurements from DNS of the turbulent periodic hill at $\mathcal{L}_z$9 with steady incompressible RANS constraints, and in the augmented version additionally enforces the Spalart–Allmaras one-equation model (Patel et al., 2023). The baseline treats Reynolds-stress divergence through a Helmholtz decomposition, whereas PINN-DA-SA splits the Reynolds forcing into an eddy-viscosity contribution plus corrective potential and solenoidal components. The network is a 7-layer MLP with 50 nodes per layer, tanh activation, 10,000 interior collocation points, and 1,000 boundary points; training proceeds with ADAM for 150,000 iterations and then L-BFGS-B for up to 300,000 iterations. At measurement spacing $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$00, PINN-DA-Baseline reduces the $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$01 mean-velocity error from $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$02 for RANS-SA to $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$03, and PINN-DA-SA further reduces it by 63% to $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$04. The reported maximum improvement over the baseline is up to 73% reduction in mean velocity reconstruction error with coarse measurements. PINN-DA-SA also achieves lower reconstruction error than variational-DA-SA across most data resolutions, except at the coarsest grid $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$05 (Patel et al., 2023).

These instantiations share the same operational goal: assimilation of sparse observations into differentiable physics while inferring hidden forcing, coefficients, or state variables. The specific unknown differs—orbital thrust, generator inertia and damping, or turbulent forcing—but the mathematical pattern is the same.

5. Sampling refinement and representation augmentation

One important branch of the PINNSR-DA idea concerns refinement of the training distribution rather than modification of the physical model. DAS-PINNs proposes a Deep Adaptive Sampling framework in which the PDE residual is treated as a probability density and approximated by a normalizing-flow model called KRnet (Tang et al., 2021). The paper provides two variants: DAS-R, which replaces interior collocation points by samples drawn from the residual-induced density and uses importance weighting, and DAS-G, which augments the existing training set with new residual-driven samples. The residual-induced target density is

$$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$06

with $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$07 used explicitly. KRnet is trained by minimizing a cross-entropy approximation to $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$08, and new points are sampled by inverse mapping from a simple latent prior. On low-regularity 2D elliptic problems and 10D linear and nonlinear PDEs, DAS-R and DAS-G substantially outperform uniform sampling and a heuristic residual-based refinement baseline. The paper states that the methods can significantly improve accuracy, especially for low regularity and high-dimensional problems, and proves monotone decrease of the expected residual loss under ideal density matching (Tang et al., 2021).

A second branch modifies the neural representation by augmenting the input dimension. DaPINN replaces the original coordinates by feature-augmented inputs such as power series terms, Fourier features, or replicas, and rewrites the residual through chain-rule-consistent derivatives in the augmented coordinates (Guan et al., 2022). The network architecture remains fully connected, with tanh activation and Adam at learning rate $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$09, but the input may change from $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$10 to vectors such as $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$11, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$12, or $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$13. Across forward and inverse benchmarks, the paper reports that the error of DaPINN is 1–2 orders of magnitude lower than that of PINN in most experiments. Representative numbers include a diffusion-reaction case where, with 65 points, standard PINN has $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$14 relative error while DaPINNs are near $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$15, and a Burgers benchmark where third-order power-series augmentation yields errors one order of magnitude lower than PINN in steep regions (Guan et al., 2022).

These works are not identical to the granular PINNSR-DA formulation. Rather, they show that the “SR-DA” component can also be interpreted as sampling refinement or dimension augmentation, provided the goal remains physics-constrained reconstruction from limited or difficult data.

6. Sparse regression and symbolic interpretability

Interpretability is a central motive in the explicit PINNSR-DA formulation and in adjacent symbolic-regression pipelines. In the granular study, sparse regression acts directly on a hand-built candidate library and yields a three-term governing equation whose coefficients are later organized by dimensional analysis (Han et al., 18 Sep 2025). This is equation discovery in the strict sense: the neural network is used to denoise and differentiate, but the final output is a compact ODE.

A different route to interpretability is provided by symbolic regression for PDE solutions using pruned differentiable programs. The workflow first trains a PINN to solve the target PDE, then generates a dense supervised dataset from the trained surrogate, then fits a Differentiable Program Architecture constrained by a context-free grammar, and finally prunes the resulting symbolic program in a depth-first manner (Majumdar et al., 2023). The grammar includes unary operators $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$16, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$17, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$18, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$19, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$20, binary operators $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$21 and $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$22, variables such as $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$23, $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$24, and $$\mathbf{a}(t)=f(\mathbf{x},t)+g_\theta(t,\mathbf{x})$$25, and constants. Each output field has its own DPA instance. Across diffusion, Kovasznay flow, Taylor–Green vortex, diffusion–reaction, and an industrial air-preheater case, pruning reduces DPA parameters by an average of 95.3% and improves accuracy by an average of 7.81% relative to the unpruned DPA, while maintaining accuracy at par with PINNs (Majumdar et al., 2023).

This suggests two distinct but compatible meanings of sparsity within the PINNSR-DA landscape. In one meaning, sparsity refers to selection of active terms in a governing equation, as in STRidge-based discovery for granular rheology. In the other, it refers to pruning a symbolic representation of a learned solution or latent operator. Both approaches turn a differentiable neural surrogate into a more explicit mathematical object.

7. Limitations, assumptions, and extensions

The cited works identify several recurrent limitations. In satellite SR-DA, omitted perturbations in the known physics model can be absorbed by the learned anomalous acceleration, which blurs physical interpretation unless priors or constraints are imposed; angles-only geometry also induces degeneracies, and sparse or noisy data enlarge the admissible solution manifold (Varey et al., 2024). In power-system identification, slow dynamics and poor excitation create flat objective surfaces that hinder identifiability and slow convergence, and computational cost remains higher than UKF despite good performance in low-inertia fast dynamics (Stiasny et al., 2020). In turbulence reconstruction, the baseline RANS formulation is underdetermined away from measurements, SA augmentation inherits the limitations of a linear eddy-viscosity closure, and at the coarsest data resolution the variational method can be more robust than the PINN (Patel et al., 2023).

Method-specific caveats also matter. DAS-PINNs adds the cost of training a generative model and can suffer if the learned sampling density becomes too concentrated (Tang et al., 2021). DaPINN depends on appropriate feature choice: mismatched augmentations can concentrate error rather than reduce it (Guan et al., 2022). The granular PINNSR-DA framework depends on a sufficiently rich candidate library, requires multiple datasets across parameter ranges for the dimensional-learning stage, and incurs higher computational cost than classical SINDy (Han et al., 18 Sep 2025). The differentiable-program symbolic-regression pipeline relies on a greedy pruning rule and a fixed grammar that may be suboptimal for some PDEs (Majumdar et al., 2023).

The extension paths reported in the literature are correspondingly diverse. Satellite SR-DA proposes multi-satellite joint estimation, maneuver detection and segmentation, multi-sensor fusion, hybrid filters, and uncertainty quantification (Varey et al., 2024). The turbulence paper highlights multi-equation closures, uncertainty quantification, multi-fidelity assimilation, hybrid PINN-plus-discretization strategies, and domain decomposition for 3D scaling (Patel et al., 2023). The granular framework is proposed as extensible to jamming, segregation, and fluid–particle interactions (Han et al., 18 Sep 2025). A plausible implication is that PINNSR-DA is not a single endpoint method but a modular design pattern: physics-informed learning supplies differentiable structure, while sparse regression, dimensional analysis, adaptive sampling, feature augmentation, or symbolic post-processing determine whether the final product is a reconstructed state, a learned forcing, or a compact governing law.