NewtonGen: Adaptive Optimization & Video

• NewtonGen is a dual-framework that integrates a Hessian-informed optimizer for dynamic learning rate selection with a physics-based text-to-video generator.
• The optimization component uses a local quadratic approximation to compute optimal step sizes, significantly reducing hyperparameter tuning overhead.
• The video generation module leverages latent neural ODEs to enforce Newtonian dynamics, resulting in controllable and realistic text-to-video outputs.

NewtonGen refers to two distinct but technically advanced contributions in the fields of optimization and generative modeling: (1) an adaptive optimization algorithm—generalized Newton’s method (GeN)—for dynamically selecting learning rates in gradient-based algorithms, and (2) a framework for generating text-to-video content with physics-consistent and controllable dynamics via trained latent neural ODEs, known as Neural Newtonian Dynamics (NND). Both usages are grounded in Newtonian principles and employ data-efficient trainable systems to improve either convergence efficiency (in optimization) or physical realism and controllability (in generative modeling) (Bu et al., 2024, Yuan et al., 25 Sep 2025).

1. Generalized Newton’s Method (GeN) for Optimization

NewtonGen (GeN) is a Hessian-informed learning-rate selection method that generalizes the Newton–Raphson update to arbitrary optimizers. For a given preconditioned gradient descent step: $w_{t+1} = w_t – η_t\,g_t^{\text{optim}},$ where $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$ for a preconditioner $P_t$, GeN leverages a local quadratic approximation to the loss: $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$ yielding a locally optimal step length: $η_t^* = \frac{G^⊤g}{g^⊤H g},$ with $G = ∇L(w_t)$, $H = ∇^2L(w_t)$, and $g$ as above. This optimal $\eta_t^*$ maximizes the predicted decrease in loss for any optimizer direction (Bu et al., 2024).

When $g = H^{-1}G$, the update reduces to the classical Newton–Raphson step, highlighting GeN’s generality. For SGD and Adam, the approach only requires an inexpensive additional pair of no_grad forward passes every $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$0 iterations for computing finite-difference loss values, significantly reducing the cost of hyperparameter tuning and learning rate scheduling.

2. NewtonGen for Physics-Consistent Text-to-Video Synthesis

In the domain of generative video modeling, NewtonGen is a system for producing physically consistent, controllable text-to-video outputs by integrating learnable Newtonian dynamics with a data-driven video diffusion backbone (Yuan et al., 25 Sep 2025). NewtonGen addresses limitations of prior text-to-video models, such as nonphysical motion and poor parameter control, by injecting structured dynamical constraints via a Neural Newtonian Dynamics (NND) module.

NewtonGen’s architecture comprises two stages:

• Dynamics Learning (NND): Trains a neural ODE with both linear second-order ODE terms and a residual MLP on a small, clean, simulated video dataset. Latent physical state vectors $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$1 encode position, velocity, rotation, angular speed, size, and area. The NND is supervised via L2 loss against encoder outputs $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$2.
• Controllable Generation: Parses the user’s text for initial conditions and physical parameters, runs the ODE solver to output latent trajectories, projects these to pixel-level optical flows, and conditions a diffusion video generator on both text and structured flows. This enforces that generated videos respect the underlying Newtonian dynamics.

3. Mathematical Formulations and Algorithms

NewtonGen (GeN) Update Rule

Given a loss function $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$3, gradient $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$4, Hessian $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$5, and optimizer direction $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$6, the locally optimal step size is

$g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$7

with parameter update

$g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$8

Finite-difference estimation for $g_t^{\text{optim}} = P_t^{-1}∇L(w_t)$9 uses three loss evaluations: $P_t$0 with momentum-based smoothing: $P_t$1

Neural Newtonian Dynamics (NND)

For latent state $P_t$2, the ODE is structured as: $P_t$3 for each coordinate $P_t$4, solved via continuous integration: $P_t$5 where $P_t$6 returns first and second derivatives according to the learned ODE and residual MLP.

4. Implementation, Training, and Efficiency

NewtonGen (GeN) for Optimization

Implementation requires only two additional forward passes every $P_t$7 steps, which, given typical $P_t$8 (backward vs. forward cost), results in ≈8% computational slowdown for $P_t$9, with essentially unchanged memory footprint. Three mini-batch loss values yield a quadratic fit for $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$0. Momentum smoothing stabilizes learning rate adaptation. GeN supports any optimizer; for Adam/AdamW, $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$1 is the preconditioned gradient.

NewtonGen for Generation

• Video-Physical Encoder: Employs SAM2 segmentation and OpenCV-based feature extraction.
• NND Training: AdamW optimizer, cosine annealing, trained on batches of simulated “physics-clean” videos across 12 motion types. Residual MLPs correct for nonlinear or unmodeled effects (crucial for motions such as parabolic+rotation).
• Controllability: Text parsing extracts physical initial conditions directly mapped to latent ODE state $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$2. Optical flow projections enforce latent motion within the diffusion space, ensuring realism and fine-grained user control.

5. Empirical Evaluation and Results

GeN Optimizer

Across domains (toy nonlinear optimization, image classification, language tasks, object detection, segmentation, and generative modeling), GeN-equipped optimizers attain or exceed the accuracy of heavily tuned learning-rate schedules, without requiring custom schedulers, warmup, or decay schemes. For example:

• ResNet50/CIFAR-10 (5 epochs): GeN–SGD: 96.8%, versus SGD(constant) 95.9% and SGD(cosine) 95.8%.
• GPT-2 LoRA/E2E NLG: BLEU: GeN=67.30, versus linear decay 66.85, cosine 66.59.
• Mask R-CNN/Penn-Fudan: Detection AP: GeN=0.805, manual LR 0.802; Segmentation AP: GeN=0.771, manual 0.768.

NewtonGen for Text-to-Video

On 12 distinct Newtonian motions, NewtonGen achieves best physical invariance scores (PIS) on 11/12 categories and near-simulator reference on the twelfth, outperforming SORA, Veo3, CogVideoX-5B, Wan2.2, and PhysT2V. Video outputs exhibit:

• Gravity-consistent arcs, constant-speed slides, orbital rotation, realistic 3D approach, natural deformation.
• Precise, immediate control in response to altered initial velocities, directions, and world settings, due to direct ODE conditioning. Ablation studies indicate strong dependence on both the residual MLP (for nonlinear dynamics) and modest training data (100 videos per motion type yield near-maximal gains).

6. Theoretical Insights, Limitations, and Directions

GeN (Optimization)

• Local Quadratic Optimality: The GeN step maximizes the decrease in loss along the optimizer direction under the second-order Taylor expansion.
• Scale Invariance and Robustness: Updates are invariant to rescaling the optimizer direction, and maintain descent guarantees when $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$3.
• Data/Noise Robustness: Batch estimation noise in $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$4 is $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$5 with fitting error $$L(w_t) – L(w_t–η g) ≈ η G^⊤g – \frac{1}{2}η^2 g^⊤H g,$$6; large batches reduce variance.

NewtonGen (Video)

• Disentanglement: Decouples appearance from dynamics, supporting efficient improvement of realism without end-to-end retraining of diffusion modules.
• Latent Dynamics Efficiency: ODE integration over low-dimensional state is computationally efficient and suitable for real-time control.
• Limitations: Only continuous single-object Newtonian dynamics are modeled; collisions, multi-object interactions, impact phenomena, and higher-order (e.g., fluid, soft-body) effects are outside current scope. Multi-object scenarios may be approximated by running independent NNDs, with some degradation in compositional realism.

Ongoing work aims to expand NND expressivity (event-driven dynamics, fluid and soft-body modeling) and shift toward fully end-to-end physics-aware video generators (Yuan et al., 25 Sep 2025).

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References:

• "Gradient descent with generalized Newton's method" (Bu et al., 2024)
• "NewtonGen: Physics-Consistent and Controllable Text-to-Video Generation via Neural Newtonian Dynamics" (Yuan et al., 25 Sep 2025)