Multi-Modal Scientific Discovery:
Spatial, Relational, and Hierarchical Data
via the Causal Dynamics Engine

1Introduction

Our companion paper demonstrated ARDA's ability to recover causal structure from temporal observations alone across four real-world datasets. However, real scientific data is rarely uni-modal. Molecular dynamics trajectories carry spatial coordinates, network neuroscience data encodes structural connectivity, and biological systems operate across hierarchical scales from molecules to cells to organisms.

This paper asks a specific question: does providing ARDA with structural information alongside temporal observations improve causal discovery, and if so, by how much? We design four controlled ablation experiments, each comparing CDE performance with multi-modal input against a temporal-only baseline using identical dynamical data.

Our contributions:

• i.First systematic ablation of multi-modal vs. temporal-only causal discovery within a single platform.
• ii.2.7× reduction in causal ambiguity when spatial + graph structure is provided for spring-mass dynamics.
• iii.Demonstration of ARDA's scientific integrity: no false improvement when extra modalities carry no information.
• iv.Implementation and validation of a novel hierarchy-aware pooling encoder for multi-scale systems.
• v.Bug fixes to ARDA's data profiler and model selector deployed during this campaign, improving robustness for particle systems.

2Multi-Modal Architecture

2.1 Data Schema

ARDA's Episode schema natively supports five data modalities. Each modality triggers automatic selection of specialized neural encoders through ARDA's automatic profiling pipeline.

Table 1: Data modalities and their automatically selected encoders.

2.2 Encoder Composition

ARDA composes modality-specific encoders into a unified representation. Each encoder produces embeddings that are fused before the dynamics model.

Key architectural decisions validated during this campaign:

Table 2: Encoder selection logic refined during this campaign.

2.3 Bug Fixes Deployed

This campaign exposed two bugs in ARDA's automatic module selection, both fixed and deployed:

3Experimental Design

3.1 Ablation Protocol

Each experiment follows a paired ablation design: the same dynamical system is submitted to ARDA twice — once with full multi-modal input and once with temporal observations only. Both runs use identical CDE configuration, hardware, and Truth Dial (Validate). The only difference is the presence or absence of structural priors (spatial coordinates, graph edges, or hierarchy mappings).

3.2 Datasets

Table 3: Overview of ablation experiments.

3.3 Metrics

We report five primary metrics for each CDE run:

Table 4: Primary evaluation metrics.

3.4 Compute Infrastructure

All experiments executed on an NVIDIA T4 GPU (16 GB VRAM) via Hugging Face Spaces (farguney/arda-gpu). ARDA v0.1.0, Python 3.11, PyTorch 2.10 (CUDA 12.1). Worker timeout: 1800s. All runs use the Validate Truth Dial with CDE mode.

4Experiment 1: Spring-Mass Particle Network

4.1 System Description

A network of 5 point masses connected by 4 springs in a linear chain (1–2–3–4–5). Each particle has 2D position and velocity (4 state variables per particle, 20 total observables). Springs follow Hooke's law with stiffness k = 1.0 and equilibrium length r₀ = 1.0. Integrated with RK4 at dt = 0.01s for 100 timesteps from 6 random initial conditions.

The multi-modal condition provides: observations [T=100, D=20], spatial_coordinates [T=100, N=5, d=2], and graph_edges [[0,1],[1,2],[2,3],[3,4]]. The temporal-only condition provides only observations [T=100, D=20].

4.2 Results

4.3 Analysis

This is the headline result. Both conditions achieve identical path fidelity (0.994) — the CDE can reconstruct the trajectories equally well either way. But the multi-modal condition has 2.7× lower causal ambiguity (0.268 vs. 0.716), recovers all 4 ground-truth spring connections (vs. zero), and achieves a theory score of 0.99 vs. 0.84. The confidence system classifies the multi-modal result as “high / strong”and the temporal-only result as “low / insufficient.”

The implication is profound: the same data, the same physics, the same compute — but providing spatial coordinates and graph topology transforms the output from scientifically unusable to publication-ready. ARDA does not just reconstruct dynamics; with structural priors, it identifies which interactions producewhich effects.

5Experiment 2: Kuramoto Coupled Oscillators

5.1 System Description

Eight phase oscillators coupled on a ring graph with nearest-neighbor coupling (K = 2.0). The state is the set of phases θ₁, …, θ₈ governed by the Kuramoto model:

Natural frequencies ω_i drawn from N(1.0, 0.3). The with-graph condition provides the ring adjacency as graph_edges; the without-graph condition provides only phase observations.

5.2 Results

5.3 Analysis

No measurable difference. Both conditions achieve near-zero CDE ambiguity (7×10⁻⁶), identical path fidelity (~0.952), and identical “high / strong” classification. The sinusoidal coupling in the Kuramoto model is simple enough that CDE fully resolves the causal structure from phase dynamics alone. The graph input provides no additional constraint.

This is an important negative control: ARDA does not blindly exploit structural hints to inflate metrics. When the temporal signal is sufficient, additional modalities produce no artificial improvement. This demonstrates scientific honesty in the platform's multi-modal fusion.

6Experiment 3: Lennard-Jones 3-Body Molecular Dynamics

6.1 System Description

Three particles interacting via the Lennard-Jones (12-6) potential — the standard model for van der Waals interactions in molecular dynamics:

Parameters: ε = 1.0, σ = 1.0. Each particle has 2D position and velocity (12 observables total). Integrated with velocity Verlet at dt = 0.001 for 200 timesteps from 6 random initial conditions with minimum separation constraints.

6.2 Results

6.3 Analysis

Again, no measurable difference. With only 3 particles in a fully-connected topology (every particle interacts with every other particle), there is no structural ambiguity for the graph to resolve. The CDE correctly identifies that the complete graph is the only possible topology for a 3-body fully-interacting system.

This result carries a specific physical insight: Lennard-Jones interactions are pairwise and symmetric. In a 3-body system, the interaction graph is trivially complete — there is only one possible graph. Providing it explicitly gives the CDE no new information. For larger molecular systems (N > 10), where the effective interaction graph is sparse (cutoff-dependent), we predict spatial + graph input would show improvement analogous to the spring-mass result.

7Experiment 4: Hierarchy-Aware Pooling

7.1 System Description

A synthetic two-level hierarchical system: 6 oscillating entities grouped into 2 subsystems of 3 entities each. Each subsystem has internal coupling (k_intra = 2.0) while inter-subsystem coupling is weaker (k_inter = 0.3). The hierarchy mapping is:

The with-hierarchy condition provides the hierarchy_mappings dictionary. The without-hierarchy condition provides only temporal observations. This experiment also validates the newly implemented HierarchyAwarePooling encoder.

7.2 Results

7.3 Analysis

A modest but measurable improvement: 25% lower CDE ambiguity (0.113 vs. 0.151) and lower graph entropy (0.452 vs. 0.604) when the hierarchy mapping is provided. Both conditions reach “high / strong” classification, but the hierarchy-aware version produces a cleaner, more structured causal graph.

This validates the end-to-end implementation of HierarchyAwarePooling: from schema definition through data profiling, tensor extraction, batching, and encoder forward pass. The encoder correctly pools entity features within groups at each hierarchical level, producing multi-scale representations that reduce the dynamics model's uncertainty about which entities interact.

8Cross-Experiment Analysis

8.1 Summary Table

Table 5: Complete ablation results across all experiments and conditions.

8.2 Key Findings

8.3 When Does Multi-Modal Input Help?

The pattern across experiments is clear: multi-modal input helps when and only when the additional modality provides information the temporal signal alone cannot resolve:

Table 6: Multi-modal input helps precisely when structural information reduces causal search space.

9Discussion

9.1 Implications for Product

These results directly inform ARDA's product positioning:

9.2 Limitations

9.3 Future Work

Three directions emerge from this study:

• i.MD17 / rMD17 molecular benchmark: Apply ARDA to the standard molecular dynamics benchmark with real DFT-computed forces and energies.
• ii.Large-N scaling study: Test spring-mass and Kuramoto systems at N = 50, 100, 500 to establish scaling behavior of multi-modal improvement.
• iii.Dynamic graph ablation: Time-varying contact networks (e.g., epidemic models) where graph_dynamic_edges captures evolving topology.

10Reproducibility Protocol

All experiments are reproducible via ARDA's REST API.

10.1 Run IDs

Table 7: Run IDs. Retrieve via GET /v1/runs/{run_id}/result.

10.2 Multi-Modal API Usage

POST https://farguney-arda-gpu.hf.space/v1/discover
Headers: X-API-Key: YOUR_KEY, Content-Type: application/json
Body: {
  "episodes": [{
    "timestamps": [0.0, 0.01, 0.02, ...],
    "observations": [[x1,y1,vx1,vy1, ...], ...],
    "spatial_coordinates": [[[x1,y1],[x2,y2],...], ...],
    "graph_edges": [[0,1],[1,2],[2,3],[3,4]],
    "hierarchy_mappings": {
      "subsystem": [0, 0, 0, 1, 1, 1]
    }
  }],
  "mode": "cde",
  "config": {"truth_dial": "validate"},
  "project_id": "PROJECT_ID"
}

11Related Work

Equivariant GNNs (Satorras et al. 2021) [1]: E(n)-equivariant graph neural networks for particle systems. ARDA uses equivariant spatial encoding for non-grid particle data, selected automatically.

NRI (Kipf et al. 2018) [2]: Neural relational inference for interacting systems. ARDA's CDE extends NRI's graph learning with continuous dynamics and calibrated edge posteriors.

GNS (Sanchez-Gonzalez et al. 2020) [3]: Graph network simulators for particle-based physics. Unlike GNS which focuses on forward simulation, ARDA's CDE performs inverse causal discovery.

Directional message passing networks (Gasteiger et al. 2020; Schütt et al. 2018) [4, 5]: Equivariant architectures for molecular property prediction. Future work could incorporate these as alternative spatial encoders for molecular data.

Kuramoto Model (Kuramoto 1984) [6]: Canonical model for synchronization in coupled oscillator networks, widely used in neuroscience, power systems, and social dynamics.

Lennard-Jones Potential [7]: Standard pairwise potential for molecular dynamics, modeling van der Waals interactions. Parameters (ε, σ) determine the equilibrium distance and well depth.

12Conclusion

We have presented the first systematic ablation study of multi-modal input for autonomous scientific discovery, demonstrating three key findings:

• i.Multi-modal input (spatial + graph) reduces causal ambiguity by 2.7× on spring-mass particle networks, transforming results from “insufficient” to “strong” scientific usefulness and recovering all ground-truth causal edges.
• ii.ARDA demonstrates scientific integrity: when additional modalities carry no information (Kuramoto oscillators, Lennard-Jones 3-body), results are identical with or without structural priors.
• iii.A new hierarchy-aware pooling encoder reduces causal ambiguity by 25% on multi-scale systems, validated end-to-end from schema to encoder output.

These results establish that ARDA is not merely a time-series analysis tool — it is a genuinely multi-modal scientific discovery platform that uses spatial, relational, and hierarchical structure to produce higher-confidence causal theories. Automatic encoder selection ensures scientists can provide whatever data they have without needing to understand the underlying architectures.

References

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