Stochastic Inverse Physics-Discovery Framework

• The SIP Framework is a suite of computational and statistical methodologies that infers governing physical laws from high-dimensional, noisy data by treating model parameters probabilistically.
• It employs Bayesian inference, sparse identification, and physics-informed neural networks to ensure physical consistency and robust uncertainty quantification.
• The framework has been successfully applied in fields like climate modeling and biological networks, demonstrating significant improvements in predictive accuracy and model interpretability.

The Stochastic Inverse Physics-Discovery (SIP) Framework encompasses a suite of computational, statistical, and machine learning methodologies for uncovering the governing physical laws of complex systems under uncertainty. Designed to address high-dimensional, noisy, and partially observed data typical of natural and engineered systems, SIP methodologies treat coefficients, parameters, and sometimes system structure as random variables or processes, enabling simultaneous quantification of physical variability, measurement noise, and model uncertainty. SIP extends and synthesizes advances in Bayesian inference, physics-informed neural networks, generative modeling, sparse system identification, and optimization under constraints, producing interpretable, physically consistent models with well-characterized predictive confidence.

1. Core Principles and Problem Setting

The SIP framework systematically addresses the identification of governing equations for systems described by stochastic differential equations (SDEs), stochastic partial differential equations (SPDEs), ordinary differential equations (ODEs), or other physics-based models, in the presence of uncertainty in data and system parameters (2507.09740, 2410.16694). Unlike classical deterministic approaches, SIP treats key unknowns—such as coefficients in dynamical equations—as random variables or even random fields, with the goal of inferring their posterior distributions conditioned on observed data. This affords natural uncertainty quantification and enables discovering robust, generalizable models in environments characterized by system variability, unobserved forcing, or limited and noisy measurements.

The general SIP workflow involves:

• Constructing a model or library (e.g., polynomial, trigonometric, or other functional bases) relating system states and their derivatives to candidate physical laws.
• Framing the unknown model coefficients as objects to be inferred probabilistically (e.g., distributions over coefficients).
• Using Bayesian, variational, adversarial, or information-theoretic objectives (such as minimizing Kullback–Leibler divergence between push-forwarded samples and empirical data) to drive inference and model selection.
• Enforcing physical constraints, such as conservation laws or global stability, via explicit mathematical or algorithmic constraints in the inference or learning procedure (1312.1881).

2. Probabilistic Modeling and Uncertainty Quantification

A central distinction of the SIP framework is its probabilistic treatment of physical variability and epistemic (model) uncertainty. Unknown parameters (e.g., drift and diffusion coefficients in SDEs) are treated as random variables with priors reflecting sparsity, physical constraints, or prior knowledge (2507.09740). The goal is to infer a posterior over the coefficient space that, when pushed forward through the governing equations, best matches the empirical data distribution. This matching is typically formulated as minimizing the Kullback–Leibler divergence:

$$\mu^*_\Lambda = \arg\min_{\mu_\Lambda} D_{KL} \left( \hat{\mu}_Y \parallel \mu_Y \right)$$

where $\mu_\Lambda$ is the measure over coefficients, $\hat{\mu}_Y$ is the push-forward measure (the model-implied output distribution), and $\mu_Y$ is the empirical data distribution. The resulting models yield credible intervals for physically meaningful predictions and provide explicit posterior uncertainty for each inferred term or parameter.

This probabilistic consistency principle enables SIP to identify governing laws that are robust even under severe data limitations or measurement noise, and to distinguish measurement uncertainty from genuine variability in the underlying system (2507.09740, 2208.05609).

3. Methodological Building Blocks

3.1 Bayesian Inference and Sparse Identification

Bayesian frameworks with sparsity-promoting priors (e.g., spike-and-slab, Laplace, regularized horseshoe) are recurrent components of SIP algorithms (2203.11010, 2507.09740). Coefficient sparsity advances physical interpretability, selecting a minimal set of active mechanisms from a broad candidate library. This allows the inference process to recover parsimonious analytic forms of the governing equations, e.g., identifying the correct drift and diffusion terms in SDEs (2410.16694), or sparse nonlinear interactions in chaotic systems (2105.02368).

3.2 Physics-Informed Machine Learning

Physics-informed machine learning forms a foundational aspect, with their loss functions augmented by physics-based residuals (e.g., PINNs) or variational principles (1809.08327, 2008.10653). For example, loss functions combine data mismatch, PDE residuals, and regularization, and may be designed to integrate probabilistic constraints by leveraging automatic differentiation to compute derivatives required for enforcing the physical laws.

Recent advances include deep generative models with physics-informed architectures (e.g., sPI-GeM (2503.18012), PI-VEGAN (2307.11289), PI-GEA (2311.01708)) for handling highly complex, high-dimensional stochastic fields, with scalability in both stochastic and spatial dimensions.

3.3 Generative and Flow-Based Models

SIP incorporates modern generative modeling, including variational autoencoders (PI-VAE (2203.11363)), normalizing flows (NFF (2108.12956)), and score-based diffusion models with explicit score matching objectives (2301.10250), to model non-Gaussian, multimodal, and high-dimensional parameter distributions. These models enable unified treatment of forward, inverse, and mixed stochastic physics problems, often allowing explicit sampling and density evaluation for robust uncertainty quantification.

3.4 Active Learning and Experimental Design

Adaptive sensor placement and experimental design strategies, such as those guided by dropout-based uncertainty or feedback control (1809.08327, 2203.11010), are integrated into SIP workflows to maximize data informativeness in regions of high epistemic uncertainty, efficiently allocating additional measurements and perturbations to improve identification in data-scarce regimes.

3.5 Constraint Enforcement and Physical Admissibility

SIP imposes physical constraints such as energy conservation, global stability (e.g., negative definiteness of parameter-related matrices (1312.1881)), or symmetry requirements. Algorithms for constrained sampling and optimization (e.g., constrained MCMC for negative definite matrices) are central in models where physically admissible solutions occupy only a subset of parameter space.

4. Applications and Performance Evaluation

SIP frameworks have been validated across domains that include climate modeling, subsurface flow, ecological dynamics, chaotic systems, and biological networks. Representative problems include:

• Discovering governing equations from sparse, noisy, or snapshot observations without explicit knowledge of system inputs (2208.05609, 2410.16694);
• Quantifying and learning both the drift and diffusion structures in SDEs from partial or aggregate data (2008.10653, 2503.18012);
• Recovering model parameters and physical laws in systems with pronounced intrinsic or input variability, such as in the Lotka–Volterra predator–prey system, Lorenz attractor, and porous media infiltration (2507.09740);
• Large-scale, high-dimensional systems—demonstrated successful solutions for SDEs with up to 38 stochastic and 20 spatial dimensions (2503.18012).

Performance is assessed using metrics such as root-mean-square error (RMSE) of coefficients, Kullback–Leibler divergence between model- and data-implied distributions, credible interval coverage, and the accuracy and physical admissibility of discovered analytic forms. SIP methods routinely demonstrate dramatic reductions in coefficient RMSE relative to classical sparse identification (e.g., 82%–98% improvements), robust credible intervals, and the capacity to operate reliably even under considerable measurement noise and heterogeneity (2507.09740).

5. Comparison with Traditional and Contemporary Approaches

SIP distinguishes itself from deterministic discovery methods (e.g., SINDy, PySINDy, and other symbolic regression approaches) by providing physically interpretable models with quantified uncertainty, robust inference in the presence of input/noise variability, and seamless integration of physical constraints. Bayesian variants (UQ-SINDy), while providing some uncertainty quantification, typically yield wider or less accurate posteriors compared with SIP (2507.09740).

In machine learning settings, SIP frameworks employing physics-informed deep generative modeling (e.g., sPI-GeM, PI-VEGAN, PI-GEA) provide advantages in scalability, stability, and accuracy, especially for high-dimensional or partially observed systems. Domain-guided normalizing flows (NFF) offer tractable likelihoods and the ability to model highly non-Gaussian physical fields (2108.12956).

6. Extensions, Limitations, and Future Research Directions

Ongoing enhancements for SIP frameworks focus on:

• Improved disentanglement of system variability from measurement noise, enabling more precise uncertainty partitioning (2507.09740).
• Scalable, efficient sampling strategies and advanced generative modeling (e.g., importance sampling, Markov chain Monte Carlo, diffusion models) to address high-dimensional and multimodal posteriors (2507.09740, 2108.12956).
• Extension to time-dependent and multi-scale systems, allowing simultaneous modeling of temporal evolution, shocks, or discontinuities (2108.12956).
• Direct estimation of entire state distributions rather than moments, potentially leveraging variational autoencoders, score-based diffusion models, or other likelihood-free inference techniques (2109.01621, 2301.10250).
• Systematic integration of active learning and experimental design algorithms to optimize data acquisition and maximize identification efficiency in real-world experimental scenarios (1809.08327, 2203.11010).

A plausible implication is that SIP frameworks will increasingly become foundational tools in scientific disciplines requiring robust inference from noisy, uncertain, and incomplete data—expanding their reach into domains such as geophysics, systems biology, advanced manufacturing, and even gravitational wave astronomy (2108.12956).

7. Representative Mathematical Formulations

A summary of important SIP-related mathematical expressions includes:

• SDE/SDE-like system: $dX_t = f(X_t, t)dt + g(X_t, t)dB_t$
• Probability flow ODE: $$\frac{dx}{dt} = F(x,t) - \frac{1}{2} \nabla \cdot [G(x,t)G(x,t)^T] - \frac{1}{2} G(x,t)G(x,t)^T \nabla_x \log p_t(x)$$ (2410.16694)
• Score matching objective: $$J_{\mathrm{SM}}(\theta) = \frac{1}{2} \int_0^T \mathbb{E}_{x \sim p_t} \left[\|s_\theta(x,t) - \nabla_x \log p_t(x)\|^2 \right] dt$$ (2301.10250)
• KL divergence minimization for push-forward discovery: $$\min_{\mu_\Lambda} D_{\mathrm{KL}} (\hat{\mu}_Y \| \mu_Y)$$ (2507.09740)
• Sparse regression with spike-and-slab prior: $$p(\theta|Z) = \prod_k \left( Z_k \mathcal{N}(\theta_k|0, \tau^2) + (1-Z_k) \delta_0(\theta_k) \right)$$ (2208.05609)
• Losses in physics-informed neural networks and deep generative models: e.g., $$LOSS = w_\text{data}\,\mathcal{L}_\text{data} + w_\text{equ}\,\mathcal{L}_{\text{equ}} + w_{\text{bnd}}\,\mathcal{L}_{\text{bnd}}$$ (2108.12956)

8. Conclusion

The Stochastic Inverse Physics-Discovery (SIP) Framework represents an overview of statistical inference and physical modeling, providing principled, scalable, and interpretable tools for discovering governing equations in the presence of uncertainty. SIP advances model discovery beyond deterministic paradigms by integrating rigorous uncertainty quantification, domain-informed priors, and modern machine learning, enabling robust recovery of physical laws from scarce, noisy, and incomplete data. Its proven performance across canonical and real-world systems marks it as a critical methodology in the contemporary computational science toolkit.