> For the complete documentation index, see [llms.txt](https://docs.algenta.ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.algenta.ai/machine-learning/ml-sequence-cells.md).

# LSTM and GRU cells

LSTM and GRU cell forward passes with exact gate math for sequence models.

`seq_nn` provides recurrent cells (LSTM/GRU) with exact forward **and** backward math — zero Python imports. It replaces `torch.nn.LSTMCell` / `torch.nn.GRUCell` plus their autograd backward for single-step sequence models, using the standard formulations: LSTM gates i,f,g,o with `c = f*c_prev + i*g`, `h = o*tanh(c)`; GRU (Cho 2014) with candidate `n = tanh(W_n x + U_n (r*h_prev) + b_n)`, `h = z*h_prev + (1-z)*n`. It complements `nn_layers.mojo`, which has dense layers/activations but no recurrence.

| Module   | What it does                                                                                                                            |
| -------- | --------------------------------------------------------------------------------------------------------------------------------------- |
| `seq_nn` | LSTM and GRU single-step recurrent cell forward and backward passes, plus seeded Xavier parameter initialisation — zero Python imports. |

## Functions

| Function                                                                       | What it does                                                                               |
| ------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------ |
| `lstm_cell_forward(x, h_prev, c_prev, W_flat, U_flat, b, hidden)`              | One LSTM step -> `LSTMState` (h, c plus gate caches for backward).                         |
| `lstm_cell_backward(dh, dc, cache, x, h_prev, c_prev, W_flat, U_flat, hidden)` | Exact grads -> `LSTMGrads` (dx, dh\_prev, dc\_prev, dW, dU, db).                           |
| `gru_cell_forward(x, h_prev, W_flat, U_flat, b, hidden)`                       | One GRU step -> `GRUState` (h plus gate caches).                                           |
| `gru_cell_backward(dh, cache, x, h_prev, W_flat, U_flat, hidden)`              | Exact grads -> `GRUGrads` (dx, dh\_prev, dW, dU, db).                                      |
| `lstm_param_count(input_size, hidden)`                                         | Total LSTM cell parameters: `4*hidden*(input_size + hidden + 1)`.                          |
| `gru_param_count(input_size, hidden)`                                          | Total GRU cell parameters: `3*hidden*(input_size + hidden + 1)`.                           |
| `xavier_flat(rows, cols, fan_in, fan_out, seed)`                               | Seeded Xavier/Glorot-uniform flat row-major matrix; values in ±sqrt(6/(fan\_in+fan\_out)). |
| `init_lstm_params(input_size, hidden, seed)`                                   | Seeded Xavier W, U; b zero except forget-gate block = 1.0.                                 |
| `init_gru_params(input_size, hidden, seed)`                                    | Seeded Xavier W, U; b zero.                                                                |

## Layout, determinism, and failure policy

Parameters are flat row-major with gate blocks stacked i,f,g,o (LSTM) or r,z,n (GRU): W is \[n\_gates*hidden x input], U is \[n\_gates*hidden x hidden], b has length n\_gates\*hidden. Initialisation is deterministic — a seeded xorshift64 RNG means the same seed reproduces bit-identical parameters and a different seed provably differs (both asserted in the tests). `init_lstm_params` sets the forget-gate bias block to 1.0, the standard trick so early training does not forget. Invalid configurations (hidden <= 0, inconsistent sizes) return empty states/grads rather than trapping.

## Verification

Forward passes are pinned to independently hand-computed sigmoid/tanh chains at 1e-12 tolerance: the LSTM known-value case yields h = 0.20465357159216882, c = 0.4474491587609256 (all four gate activations checked too); the GRU case yields h = 0.596051095464341. Every backward output — dW, dU, db, dx, dh\_prev, dc\_prev — is checked against central finite differences (eps = 1e-5) on a 3-input/2-hidden cell with max relative error < 1e-4 required for **every** element (24 dW + 16 dU + 8 db entries for LSTM; 18 + 12 + 6 for GRU). Param-count identities are asserted exactly (`lstm_param_count(3, 2) == 48`, `gru_param_count(3, 2) == 36`), and `xavier_flat` values are verified to stay within ±sqrt(6/(fan\_in+fan\_out)).

**Next:** promote what you trained with [statistical model comparison](/machine-learning/ml-model-comparison.md), or browse the [engine catalog](/engine-runtime/catalog.md).


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