PhysisForcing: Physics-Driven Simulation & Learning

• PhysisForcing is a methodology that explicitly embeds physical forcing into models via PDE source terms, constraint forces, and energy invariants.
• It is applied across PDEs, physics-informed neural networks, and simulation frameworks to enforce realistic dynamics and improve model interpretability.
• Modular forcing techniques enable robust prediction in astrophysical, geophysical, and video generation tasks by aligning simulations with physical laws.

PhysisForcing denotes a set of methodologies that introduce explicit, structured physical forcing into the modeling, simulation, or learning of dynamical systems. The term spans a broad range of formalisms, including direct physical source terms in partial differential equations (PDEs), physics-encoded deep learning architectures, explicit constraint-force methods in PINNs, physically-driven supervision in generative video models, and force-conditioned simulation frameworks. Across these domains, the central aim is to enforce, inject, or align with physical principles—forces, constraints, or energetic invariants—rather than relying solely on data-driven or unconstrained modeling.

1. Direct Forcing in Physical PDEs and Reduced Models

PhysisForcing originated in the analysis of physically forced PDEs, notably the nonlinear Schrödinger equation (NLSE) and its generalizations. For systems like deep-water wave trains governed by the forced-damped NLSE or Dysthe equation, forcing (e.g., wind input) and damping (e.g., viscous dissipation) enter as explicit terms in the evolution equations:

$$\frac{\partial a}{\partial \xi} + i\frac{1}{2} \frac{\partial^2 a}{\partial \tau^2} + i|a|^2a = \delta_0 a + i\delta_1\frac{\partial a}{\partial \tau} + \text{(Dysthe terms)}$$

with $\delta_0$, $\delta_1$ encapsulating physical gain/loss parameters such as Miles growth rate and kinematic viscosity (Eeltink et al., 2020).

In the reduced three-wave truncation, PhysisForcing manifests as non-conservative drift in the Hamiltonian structure. Forcing ($\delta_0>0$) pulls phase-space trajectories inside the separatrix, leading to recurrence locking and potential symmetry breaking; damping ($\delta_0<0$) pushes orbits out, inducing period-doubling and phase shifts. This same structure forms the basis for interpretive analysis of modulation instability, separatrix crossing phenomena, and dynamical symmetry breaking.

Astrophysical and geophysical shear flows exhibit analogous forced-dissipative structures. For the compressible Navier–Stokes system, various forcing schemes (viscous, perturbation, and relaxation) serve to sustain large-scale shear profiles:

• Viscous forcing cancels linear viscous decay locally.
• Relaxation forcing drives the mean flow toward a target profile globally, with strength set by a timescale parameter. These enforced forces balance energy budgets and determine the system's long-time stability or instability (Witzke et al., 2016).

2. Physics Forcing in Deep Learning Architectures

A major direction for PhysisForcing is "hard" physics encoding in neural network architectures, as exemplified by PeRCNN and related frameworks (Rao et al., 2021). Here, physics is embedded coercively into the learning dynamics by:

• Precisely enforcing initial and boundary conditions via upsampling and physics-based padding
• Hard-wiring PDE terms (e.g., diffusion, advection, reaction) through fixed convolutional kernels
• Eschewing penalty-based losses for exact physics, thus removing violation of governing laws during roll-out

Nonlinearities are introduced not by standard activation functions but by explicit elementwise products (the Π-block), supporting polynomial and interaction terms up to arbitrary order. PhysisForcing in this sense yields robust, interpretable, and generalizable models, especially in sparse/noisy data regimes where weakly enforced physics (e.g., traditional PINNs) may fail.

3. Explicit Constraint-Force Methods in Physics-Informed Learning

In physics-informed neural networks (PINNs) and inverse solution reconstruction, PhysisForcing is realized via the Explicit Constraint Force Method (ECFM) (Rowan et al., 8 May 2025). Standard PINNs encode physical laws as penalty terms; if model misspecification occurs, this generates hidden "constraint forces" that depend on regularization choices and compromise interpretability.

ECFM instead introduces explicit source terms, $\lambda_i$, anchored at measurement locations:

$$\nabla\cdot \sigma(u(x)) + f(x) + \lambda(x) = 0, \quad \lambda(x) = \sum_{i=1}^C \lambda_i \gamma_i(x)$$

The optimization seeks to minimize both data misfit and the $\ell^2$-norm of $\lambda$, thus ensuring physical interpretability, robustness to loss formulation, and unique data-driven parameter identification. The explicit "forcing" can be interpreted as the minimal source needed to reconcile misspecified physics with observational data.

4. PhysisForcing in Physics-Guided Video Generation

Recent advances leverage PhysisForcing in deep generative models for video simulation and robotic world modeling. In these contexts, the term encompasses region-informed, hierarchical physical supervision (Zhang et al., 26 Jun 2026), as well as explicit conditioning on physical force signals (Gillman et al., 26 May 2025).

One approach (PhysisForcing) applies two auxiliary losses during diffusion model training:

• A pixel-level trajectory alignment loss $\mathcal{L}_{\text{pix}}$ on physics-informative (moving) regions, enforcing continuity of feature-space point trajectories
• A semantic-relational alignment loss $\delta_0$0 aligning token similarities in the generator to those in a frozen video encoder, thereby regularizing inter-object relational semantics

Force Prompting extends this, injecting force-conditioned control signals as high-dimensional tensors, encoding both local (e.g., poke) and global (e.g., wind) forces. These prompts instruct the generation process to manifest physically coherent object responses, using only visual priors and training on synthetic forced video datasets (Gillman et al., 26 May 2025).

5. Modular Forcing and Integration for Physics Simulation

PhysisForcing also refers to modular simulation architectures such as PhysGraph (Halimi et al., 2023). Here, physically meaningful forces—computed as gradients of user-specified potential energies, possibly including unseen external sources—are evaluated by an "opaque" force module. A graph neural network integrator, trained exclusively on energy reduction, learns to integrate these forces to state updates across arbitrary meshes, topologies, and physical regimes, decoupling force computation from the solution updating mechanism and achieving broad generalization.

This modularity enables seamless accommodation of new force models without retraining, as the integration module responds adaptively to any net force distribution, respecting underlying energetic or physical structure.

6. Forced Relativistic and Holographic Fluid Dynamics

PhysisForcing emerges in holographic settings as external fields (e.g., boundary scalars) sourcing explicit force terms in the conservation laws of relativistic fluids dual to gravitational systems (Wu et al., 2024):

$\delta_0$1

where $\delta_0$2 depends on gradients of the external scalar. First-order transport coefficients (shear, bulk viscosities) remain unmodified, but second-order coefficients arise solely due to these forced terms, fundamentally modifying momentum relaxation and dispersion relations.

In this context, PhysisForcing provides a controlled avenue to probe momentum relaxation, explicit translation symmetry breaking, and the interplay between hydrodynamic response and nontrivial background fields in non-conformal settings.

7. Impact, Applications, and Outlook

PhysisForcing, in its various technical realizations, provides a rigorous methodology for embedding physical structure in data-driven and simulation frameworks:

• In PDE modeling, it enables systematic study of non-conservative effects, symmetry breaking, and adiabatic invariants under forcing and damping (Eeltink et al., 2020).
• In computational astrophysics and geophysical fluid dynamics, comparative forced simulation schemes guarantee control of mean flow profiles and match energetic balances (Witzke et al., 2016).
• For learning-based simulation, PhysisForcing in GNN and deep physics-encoded networks achieves mesh- and force-independent robust prediction and control (Halimi et al., 2023, Rao et al., 2021).
• In video generation and robotic world models, physically grounded supervision leads to more reliable policy learning and closed-loop success in manipulation tasks (Zhang et al., 26 Jun 2026, Gillman et al., 26 May 2025).

A continued direction includes hybridization of data-driven and explicit-physics models, refinement of force-based conditioning interfaces, and extending region-informed physical supervision to 3D and full force-field estimation. PhysisForcing frameworks provide a principled foundation for integrating experimental observables, physical constraints, and machine learning into a cohesive modeling paradigm.