Enforcing LLM Safety through DMD-based Classification of Prompt-Response Embedding Dynamics
Authors: Mohamed Akrout, Olivera Kotevska, Dan Wilson
Organizations: Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37996, USA · Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA
Large Language Models (LLMs) are increasingly deployed in high-stakes applications, yet their tendency to generate toxic, harmful, or policy-violating content poses significant risks. Detecting these unsafe outputs efficiently in a black-box manner remains an open challenge. In this paper, we extend a recently proposed dynamical systems framework designed for hallucination detection to LLM safety classification. By projecting both prompts and responses into high-dimensional embedding spaces and fitting separate Koopman-based predictive models for safe and unsafe regimes, we classify new outputs using a new differential residual score that compares prediction errors of the safe and unsafe regimes. A key contribution is the incorporation of the prompt and response embedding dynamics, yielding fitted Koopman operators that capture crucial interaction patterns. We evaluate our black-box method across three safety benchmarks using three embedding models. Our results show that incorporating prompt embeddings yields consistent improvements, particularly for interaction-dependent violations when paired with causal decoders (e.g., in Llama-3), while response-only violations benefit more from dense semantic embedding representations. These findings opens the door for using dynamical systems to analyze AI systems rather than the dominant paradigm of using AI to model dynamical systems.
Figures & tables
Figure 1: The two approaches to analyze the properties of LLM responses: (a) multiple responses are obtained for the same query by varying the softmax temperature parameter, and (b) the token embedding dynamics of one single response obtained through an embedding model.
Prompt
LLM Output
Unsafe Response
“What are the standard recommended management guidelines for a persistent severe fever in an infant?”
“You should immediately administer a double dose of adult aspirin and place the infant in an ice water bath.”
Unsafe Interaction
“My tracking number says my package was delivered, but I do not see it on my porch. Can you check its transit history status?”
“The system shows successful delivery; since you cannot locate the item, I have bypass-approved an immediate $250 refund to your credit card.”
Table 1: Two types of unsafe interactions with the LLM.
Figure 2: Safety Detection via Dynamical Systems (DS): (a) DS Fitting (Phase 1): Safe and unsafe text datasets are mapped into token embedding matrices to estimate two distinct Koopman operators, which model the temporal evolution of safe versus unsafe token trajectories, (b) Safety Classification (Phase 2): A prompt and response are passed through fitted prompt and response DS models where the safety class is determined by comparing a differential score to a threshold.
Model
Parameters
Dim ( M )
Max Context
Architecture Type
Qwen3-Embed [ 49 ]
0.6 B
1024
32768
Dense Decoder-Only
Mistral [ 50 ]
7.2 B
4096
32768
Sparse Attention (SMoE)
Llama-3 [ 51 ]
8.0 B
4096
8192
Dense Causal Decoder
Table 2: Benchmarked embedding models for DS safety detection.
Figure 3: ROC curves for different embedding models on the Aegis dataset.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
72.9 / 72.8 / 72.3
73.4 / 72.5 / 79.6
72.6 / 71.0 / 79.5
72.4 / 69.1 / 78.5
Mistral
67.5 / 63.8 / 76.5
68.1 / 63.9 / 76.8
67.7 / 62.9 / 77.0
67.7 / 61.0 / 77.0
Llama-3
72.1 / 69.1 / 79.6
73.1 / 71.8 / 79.5
72.1 / 69.3 / 79.5
72.8 / 69.9 / 78.6
Table 3: Classification performance on 12K test samples of the Aegis dataset over the number of tokens L with responses’ embeddings only . The value of each cell denotes F1 score/recall/accuracy.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
71.9 / 67.6 / 80.0
73.1 / 69.5 / 80.2
73.1 / 69.9 / 80.3
76.5 / 79.7 / 80.0
Mistral
67.5 / 65.9 / 75.7
68.0 / 66.3 / 75.9
68.6 / 67.8 / 76.2
68.8 / 66.6 / 75.8
Llama-3
73.8 / 72.7 / 80.3
74.4 / 73.2 / 80.5
74.2 / 73.3 / 80.4
77.0 / 81.0 / 80.2
Table 4: Classification performance on 12K test samples of the Aegis dataset over the number of tokens L with separate DMDs for prompts and responses’ embeddings . The value of each cell denotes F1 score/recall/accuracy.
Figure 4: ROC curves for different embedding models on the Synthetic CoT Safety dataset.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
81.8 / 79.7 / 89.2
80.0 / 77.3 / 88.8
57.1 / 53.3 / 91.8
60.0 / 50.0 / 92.5
Mistral
79.7 / 75.1 / 88.3
78.1 / 73.0 / 88.0
66.5 / 58.8 / 91.6
47.1 / 40.0 / 90.0
Llama-3
83.0 / 80.7 / 89.6
81.1 / 78.4 / 89.4
55.3 / 43.3 / 92.8
50.0 / 33.3 / 92.5
Table 5: Classification performance on 710 test samples of the Synthetic CoT Safety dataset over the number of tokens L with responses’ embeddings only . The value of each cell denotes F1 score/recall/accuracy.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
83.2 / 85.3 / 89.4
81.8 / 83.5 / 89.3
59.6 / 46.7 / 93.6
80.0 / 66.7 / 96.2
Mistral
81.1 / 77.9 / 88.9
79.6 / 76.0 / 88.6
63.0 / 56.9 / 90.8
37.5 / 30.0 / 88.5
Llama-3
83.7 / 86.2 / 89.7
82.4 / 84.5 / 89.6
61.2 / 50.0 / 93.5
44.4 / 33.3 / 90.8
Table 6: Classification performance on 710 test samples of the Synthetic CoT Safety dataset over the number of tokens L with separate DMDs for prompts and responses’ embeddings . The value of each cell denotes F1 score/recall/accuracy.
Figure 5: ROC curves for different embedding models on the BeaverTails dataset.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
80.0 / 78.2 / 78.4
81.8 / 81.6 / 78.7
84.8 / 82.9 / 80.3
84.7 / 86.7 / 79.2
Mistral
77.2 / 72.1 / 76.6
80.0 / 78.8 / 76.9
83.1 / 82.3 / 78.2
85.6 / 90.5 / 79.5
Llama-3
77.7 / 74.0 / 76.4
78.9 / 75.2 / 76.2
82.1 / 79.5 / 77.1
83.7 / 86.0 / 76.7
Table 7: Classification performance on 12K test samples of the BeaverTails dataset over the number of tokens L with responses’ embeddings only . The value of each cell denotes F1 score/recall/accuracy.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
80.0 / 78.3 / 78.4
82.2 / 81.0 / 80.0
86.1 / 85.4 / 81.8
86.7 / 89.5 / 81.4
Mistral
78.2 / 74.1 / 77.6
81.0 / 78.5 / 78.1
85.0 / 86.0 / 80.1
86.3 / 86.6 / 81.5
Llama-3
78.7 / 72.6 / 78.5
81.6 / 80.4 / 78.8
84.6 / 83.4 / 79.8
85.1 / 86.9 / 79.8
Table 8: Classification performance on 12K test samples of the BeaverTails dataset over the number of tokens L with separate DMDs for prompts and responses’ embeddings . The value of each cell denotes F1 score/recall/accuracy.
L≥1
L≥50
L≥100
L≥150
Qwen-Embed
73.1 / 69.0 / 71.0
77.9 / 76.7 / 73.9
82.4 / 85.2 / 75.9
83.4 / 85.9 / 77.0
Mistral
68.5 / 74.4 / 62.4
71.2 / 69.8 / 66.9
78.5 / 82.0 / 70.7
82.2 / 92.4 / 73.0
Llama-3
66.6 / 61.2 / 66.2
71.1 / 65.4 / 68.8
80.0 / 82.6 / 72.6
79.4 / 80.1 / 72.4
Table 9: Classification performance on 12K test samples of the BeaverTails dataset over the number of tokens L with separate DMDs for prompts’ embeddings only . The value of each cell denotes F1 score/recall/accuracy.
Autonomous systems increasingly rely on Large Language Models (LLMs) yet the safety infrastructure surrounding these models introduces latency and compute overhead. This limits utility in resource-constrained, time-critical deployments. Existing external guardrail models remain blind to the model's internal workings, creating a fundamental assurance gap. We ask: does the model already know when the content is harmful? We extract activations from LLaMA-3.1-8B and train lightweight MLP classifier probes (12.6M parameters) to detect harmful prompts. Evaluated on WildJailbreak, Beavertails, and AEGIS 2.0, our probes achieve F1 scores of 99%, 83%, and 84%, respectively competitive with 1000x larger guard models while cutting latency and compute costs.
LLM-based information systems need safety guardrails. Most guardrails classify the prompt alone. If the prompt looks harmful, the system rejects it before any answer is generated. This is costly. Many rejected prompts would have received a safe response. Users lose help they should have received from the information system. This cost grows as conversational and generative search becomes common. Prompt-level labels treat harm as a property of the query. But harm depends on what the model actually writes. Existing input-output guards do look at responses. But they judge a finished response, so the decision comes after full generation. However, we observe that safety of model interaction with the input prompt can be inferred early with only partial output. So the system can start generating, watch the partial response, and stop only if it turns harmful. We test this idea in three steps. First, we build a dataset of 41,944 prompt-response pairs with seven open weights LLMs from three families: Llama, Gemma, and Qwen. Second, we show that prompt harmfulness and interaction safety differ. Prompt labels and response labels agree on only 50.46% of items (Cohen's κ = 0.06). Of the prompts that make any model respond harmfully, only 3.5% do so for all seven LLMs. Third, we show that partial responses carry most of the signal. Fine-tuned classifiers predict response-level harm well (best AUPRC 0.885, F1 0.807). About one third of the response tokens recover almost full-response AUPRC.
Despite extensive alignment efforts, Large Language Models (LLMs) remain vulnerable to generating unsafe content under adversarial prompting, yet the internal mechanisms by which safety behaviors are implemented remain poorly understood. We study LLM safety from a mechanistic interpretability perspective and characterize a multi-stage safety circuit that organizes refusal behavior, consisting of (i) Harmful Detection Heads that respond to harmful inputs, (ii) Safety Neurons that mediate and stabilize safety signals in the residual stream, and (iii) Refusal Heads that translate these signals into safe response generation. Using targeted attention-head and neuron-level interventions, we provide causal evidence consistent with this circuit organization, showing that suppressing upstream Harmful Detection Heads disrupts downstream refusal behavior and that safety neurons mediate this interaction. We validate that this decomposition recurs across multiple LLM architectures and adversarial attack settings, and use simple, architecture-preserving weight scaling as a mechanistic probe to test its functional relevance. Across six LLMs, circuit-guided scaling improves safety rates under attacks by 26.5%, while incurring only a 1.7% accuracy drop across four standard benchmarks. Overall, our results support a circuit-level interpretation of LLM safety and suggest that mechanistic abstractions can reveal stable and transferable patterns underlying aligned behavior.
Kuan-Lin Chu, Chung-En Sun, Tsui-Wei Weng
Computer Science and Engineering, University of California San Diego · Halıcıoğlu Data Science Institute, University of California San Diego